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AN ELEMENTARY TREATISE ON 

THE DIFFERENTIAL AND 

INTEGRAL CALCULUS, 



FOE THE USE OF COLLEGES AND SCHOOLS. 



BY G. W. HEMMING, M.A, 

FELLOW OF ST. JOHN's COLLEGE, CAMBRIDGE. 



SECOND EDITION, 
WITH CORRECTIONS AND ADDITIONS. 



Cambrtbgt : 

MACMILLAN & CO. 

1852. 



PREFACE TO THE SECOND EDITION. 

Many corrections and additions have been made in 
the present Edition, chiefly with the view of fitting it for 
younger students by bringing out the leading principles 
of the subject with greater clearness. Some additional 
illustrative examples are given, but for purposes of practice 
the student is still referred to Gregory's Collection of 
Examples. 



Lately Published, by the same Author, 12mo. cloth, 45. Qd. 

First Book on Plane Trigonometry, comprising Geometrical Tri- 
gonometry, and its Application to Surveying, with numerous 
Examples for the use of Schools. 



CONTENTS. 



CHAPTER I. 
Variables and Functions. 

Article Page 

1 — 7. Definitions of variables and functions .... 1 
8. Equations of one or more independent variables . . 7 



CHAPTER n. 

Limiting Values of Functions, 

9. Definition and explanation of a limiting value . . .8 

10. Axioms respecting limiting values . . . 11 
,, arc PQ 

''' ^^^^Och^^Q=^ '' 

12. U^'^=\ 12 

a? 

a?" ~ 1 

13. ^^,=0 7=n ....... 12 

X- 1 

14. lt,=o(l + «!y = e ...... 13 

15. «.^!^Cl±f)= » 13 

X log a 

16. /^^=o— ^ = logo ...... 14 



CHAPTER HI. 

Differentiation. 

17- Definition of differentials and differential coefl[icients . .15 

18. Differentials proportional to the rates of increase of the variables 17 

19. Geometrical illustration of the definition . . .17 
20 — 44. Differentiation of elementary functions . . . 19 
45 — 51. Differentiation of a function of a function . . .27 

52. Examples ....... 30 

53. Summary of results . . . . . .31 



VIU 



CONTENTS. 



CHAPTER IV. 

Integration. 
Article 

54. Definition of integration 

56, 57. Addition of a constant in integration 

58—65. Integration of elementary functions 

66. Summar}'^ of results 

67. Integration by algebraical transformations 
68, 69. Integration by parts 
70 — 76. Integration of rational firactions 

78. Rationalization . . i 

79, 80. Criteria of integrability of ar» (a + hr'Y dx 

81 — 87. Integration by reduction 

88. Exceptional cases 

89. Integration of sin" x cos" x dx in particular cases 
90, 91. Integration of the same function by reduction 



Page 
34 
35 
36 
40 
41 
42 
43 
49 
49 
51 
56 
57 
57 



CHAPTER V. 

Successive Differentiation. 

92. Definition of an independent variable . . 60 

93. Geometrical illustration . . . • • 61 
94, 95. Successive differentiation . . • • .62 
96 — 98. Relations between successive differentials and differential coeffi- 
cients, when the independent variable is general . 63 

99. Form of the above relations when the quantity (a?), under the 

functional sign, is independent variable . . 64 

100. Examples ..... 65 

101 — 104. Formation of differential equations . .67 

105. Homogeneity of differential equations . . .70 

106. Change of independent variable . . . .70 

107. To pass from an equation among differentials, with x as inde- 

pendent variable, to one among differential coefficients 

with respect to x, and the converse ... 70 

1 08, 109. To pass from a general independent variable to x, and the converse 7 1 

110. To pass from a general independent variable to any fimction of x 71 

111 — 113. To change the independent variable from x to any function of .r 72 

114. To change the independent variable from x to y . 74 

115,116. Illustrations and examples . . . . .74 



CHAPTER VI. 

Differentiation of Functions of several Variables and 
Implicit Functions. 

7, 118. The total differential of a function of two connected variables 
equal to the sum of the partial differentials 



78 



CONTENTS. 



IX 



Article Page 

119—121. Examples ....... 80 

122. Extension to functions of any number of variables 82 

123 — 125. Implicit functions of one independent variable . . 82 

126 — 128. Relations between corresponding implicit and explicit functions 84 

129. Examples . . . . . . .86 

130 — 132. Definition of the total differential of a function of two inde- 
pendent variables . . . . . 87 

133, 134. Another form of the definition . . . .92 

135. Geometrical illustration ..... 94 

136. Successive total and partial diflferentiation . . .95 

137. Order of successive partial differentiation immaterial . 96 

138. Notation for successive differentials and differential coeflficients 97 
139 — 142. Relations between successive total and partial differentials 

and differential coeflftcients, with general independent variables 99 

143. The same relations, with particular independent variables 101 
144 — 146. Change of independent variable in equations involving partial 

differential coefficients ..... 101 

CHAPTER VII. 

Development of Functions. 

147 — 149. Taylor's theorem. Limits of the remainder. Examples 104 

150, 151. Stirling's theorem. Examples .... 107 

152, 153. Failure of Taylor's theorem .... 108 

154. Other methods of expansion . . . .110 

155. Extension of Taylor's theorem to functions of two variables 111 

156. d-f{x,y) _ d''f{x,y) ^^^ 
dx'^dxf dy^dx^~'' 

157. Limits of remainder of the above series . . . 112 
158,159. Other forms of Taylor's theorem . . . 113 

160. "Lagrange's theorem . . . . .113 

161. Laplace's theorem . . . . . 115 

162. Laplace's theorem deduced from Lagrange's . . 116 



CHAPTER Vin. 

Limiting values of Indeterminate Functions. Maxima and Minima 
values of Functions. 
Fractions of the form ^ . 



163. 

164,165. 

166. 

167—171. 

172—177. 

178. 



Fractions of the form ■§■ . 

Other indeterminate forms .... 

Maxima and Mimma of Functions of one independent variable. 

Geometrical illustration . . . 

Maxima and Minima of Functions of two or more independent 

variables. Geometrical illustration . 
Examples ....... 



117 

117 
119 

119 

123 
129 





CONTENTS. 








CHAPTER IX. 








Tangents and Asymptotes. 






Article 






Page 


179. 


Definition of a tangent 


. 


133 


30, 181. 


Equations to tangent and normal, dx^ + dy^ - ds^ 




133 


182. 


Length of tangent and normal 




135 


183. 


Concavity and convexity of curves 




135 


184. 


Points of inflexion .... 


. 


136 


185. 


Asymptotes ..... 




136 


186. 


Examples ..... 




137 


187. 


Inclination of tangent to radius vector, dr^ + rVe^ = 


-ds' . 


140 


188. 


Values of 5T and SF .... 


. 


141 


189. 


Asymptotes of polar curves 




143 


190. 


Concavity and convexity of polar curves. Points of inflexion 


143 



CHAPTER X. 

Contact of Curves. 

191, 192. Definition of contact. Curves whose contact is of a higher 
order, indefinitely closer than those whose contact is of a lower 

order ...... 144 

193. Curves whose contact is of an even order cut, otherwise Jiot 145 

194. Conditions of contact modified . . . 145 

195. Curve of closest contact . , . . . , 146 
196 Circle of curvature ..... 147 
197. Radius of curvature and coordinates of centre of curvature 147 

198,199. Various forms of these expressions . . . 149 

200. Circle of curvature has a contact of third order when the radius 

is a maximum or minimum .... iSl 

201. Equation to evolute . . . . . 152 
202, 203. Properties of evolute . . . . .153 

204. Evolute of polar curves . . . . 156 







CHAPTER XI. 






Singular Points. 


205. 


Definitions 


. 


206—208. 


Multiple points 


. 


209. 


Cusps 


. 


210. 


Conjugate points 


. 


211. 


Other singular points 


212. 


Summary. Example . 



157 
157 
161 
162 
164 
164 



CONTENTS. 



Article 

213, 214. Explicit Curves 

215. Implicit Curves 

216. Polar Curves 



CHAPTER XII. 

Tracing of Curves. 



Page 
166 
167 
170 



217,21! 
219. 

220. 

221. 
222. 
223. 
224. 



CHAPTER XIII. 

Envelopes — Differentials of Areas, Sfc. 

Envelope of a group of curves . 

Ultimate intersection of contiguous normals 

dr 

dp 
Differential of an area 
Differential of a sectorial area 
Differential of the volume of a solid of revolution 
Differential of the surface of a solid of revolution 



172 
175 

175 

176 
176 
177 
177 



CHAPTER XIV. 

Integration between Ldmits. Successive Integration. 

225. Integral the limit of the sum of a series 

226. Area of a curve .... 

227. Length of^a curve 

228. Sectorial area jof a curve . 

229. Volume and surface of a solid of revolution 

230. The same results differently obtained 

231. Further application of Art. 225 
232 — 238. Successive integration. Examples 

239. Integral expressed by a series . 

240. Remainder of Taylor's series as a definite integral 



178 


180 


181 


182 


182 


182 


184 


185 


192 


ral . 192 



DIFFERENTIAL AND INTEGRAL CALCULUS. 
CHAPTER I. 

VARIABLES AND FUNCTIONS. 

1. In all analytical inquiries the symbols with which we are 
concerned are divisible into two classes — those which by the con- 
ditions of the problem admit of variation, and those which are in- 
capable of it. The former are called variables, the latter constants. 
Thus, the co-ordinates of any point of a given curve are variables, 
"being capable of a change of value in passing from one point of the 
curve to another, while those quantities which enter into the equa- 
tion to determine the particular curve of which we treat, as the 
latus rectum of a parabola, the semi-axes of an ellipse, the radius of 
a circle, &c., are constants so long as the equation is restricted to 
one particular curve. 

It should however be borne in mind, that the same quantities 
may appear in one problem as variables and in another as constants. 
Thus, if ^r and y represent the co-ordinates of any point in a given 
circle, they are connected by the equation 

where x and y are variables and r constant. But if we have to 
determine the locus of the intersection of pairs of equal circles, 
having their centres in two fixed points whose distance is 2&, the 
equations to any pair of circles whose radii equal r may be put in 
the form 

{x-hyA-f^T\ 

{x + hY + y^ = r\ 

The equation to the required locus will then be a relation between 
X and y, which holds for every possible value of r and will be found 
from the above equations by treating r as a variable. The distance 
26, between the centres is the same for every pair of circles. In this 
problem therefore x, y and r are variables and b constant. By 
H. D. 0. 1 



2 VARIABLES AND FUNCTIONS. 

eliminating r between them we find the equation to the locus 
required, viz. x — Q. 

Hence we obtain the following definitions. 

A variable is a quantity which, by the conditions of the problem 
that we are investigating, may assume a number of different values. 

A constant is a quantity which the conditions of the problem 
suppose to be invariable. 

Quantities, which in some problems appear as variables and in 
others as constants, as the radius of the circle in the above example, 
are sometimes termed parameters. 

2. A continuous variable is one which is capable of receiving 
any value whatever ; a discontinuous variable, one which only ad- 
mits of a particular series of values separated by finite intervals. 
Thus, the co-ordinates of an indefinite straight line are continuous 
variables, while a symbol which represents any integer is a discon- 
tinuous variable. 

In the ordinary operations of Algebra and in the present subject, 
all the variables employed are supposed to be continuous. 

Results, however, obtained on the supposition of continuity are 
applicable to variables which are continuous within certain finite 
limits, and become discontinuous beyond them. Thus, the general 
equation to a straight line is applicable to the side of a triangle, 
provided we restrict the values of the co-ordinates to the limits 
imposed by the figure. 

Variables are usually represented by the final letters of the 
alphabet, as u, v, x, y, z, &c., and constants by the initial letters a, 
hj c, &c. ; a notation purely arbitrary, and adopted for convenience. 

3. A function of any variable is a quantity whose value depends 
upon and changes with that of the variable, so that when an arbi- 
trary value is given to the variable the function receives a definite 
corresponding value. 

Thus, the distance which a man walks in a given time is a func- 
tion of his rate of walking ; and the algebraical expressions a^, a', 
a + hx^'i sin x^ Sec, are all functions of x, and x a function of each of 
them. So also, the co-ordinates of a curve are mutually functions 
of each other, since when an arbitrary value is given to one of them, 
the other receives a definite corresponding value. 



VARIABLES AND FUNCTIONS. 3 

Functions of the variables x, y, &c. are writteny*(.r), f{y), <p (x)^ 

When the same functional sign is prefixed to different variables, 
it denotes that the function is obtained from the variables in the 
same way. Thus, if/(x) = sin x, /(z), if it occurs in the same in- 
vestigation, must = sin ^. 

4. A function of a continuous variable is said to be continuous 
when its values are obtained from those of the variable % a Jixed 
law, such that a gradual change in the variable produces a corre- 
sponding gradual change in the function, otherwise it is discon- 
tinuous. The meaning of these two conditions of continuity becomes 
-more apparent by the aid of a geometrical illustration. 

Suppose a curve to be traced whose equation is 1/ =f(^x). Such 
a curve must always exist whatever be the form of/, since we can 
.fix as many points of it as we please by giving a series of arbitrary 
values to x, and determining from the equation the corresponding 
values of ^. In order that the ordinate of this curve may be a con- 
tinuous function of the abscissa, according to the above definition, 
two conditions must be satisfied : 

(1) The change of^ consequent on a gradual change o£ x, must 
be gradual. ^ 

(2) It must folloAV^ a fixed law. 

The former of these conditions excludes the case of a series of 
disconnected points, or of lines terminating abruptly, and restricts 
us to curves which may be traced without raising the pen from the 
paper. Where however ^ has several values corresponding to each 
value of X, the condition is satisfied if each branch may be thus con- 
tinuously drawn. The latter condition excludes from the class of 
continuous curves, figures composed partly of one curve and partly 
of another, as for instance a triangle, the co-ordinates of whose 
several sides are related by three distinct laws or equations. It is 
even possible to trace curves which satisfy the geometrical idea of 
continuity whose ordinates are nevertheless discontinuous functions 
of the abscissae in the sense of our analytical definition. 

A curve is considered continuous in geometry where it can be 
;traced by the motion of a point, without any abrupt changes of 
direction (Newt. Frin, Bk. i. Lem. 5). A triangle is therefore geo- 

1—2 



4 VABIABLES AND FUNCTIONS. 

metrically discontinuous, but the egg-shaped curve formed by con- 
necting the extremities of a semi-circle with those of a semi-ellipse 
of equal diameter satisfies the geometrical definition. The equation 
to the latter figure however does not give ?/ as a continuous function 
of X, since for some values of the abscissa it follows the law of the 
circle and for others that of the ellipse. But the equations to the 
distinct parts of such composite figures are continuous, and results 
obtained by treating them as such may be applied to the geometrical 
figure if the values given to the variables are confined within the 
limits appropriate to the particular equation employed. Thus in 
the last example the results deduced from the continuous equation 
to the circle are true of the semi-circular portion of the figure. 

In the above instances of discontinuity the change in value of 
the function was gradual, but the law was not fixed. In other cases 
the law remains fixed, but there is an abrupt change in value at 
particular points. Discontinuity of the latter kind occurs most fre- 
quently in the form of a sudden change from +co to -co. Thus 

for example, when x passes through the value - , sec x changes 

abruptly from +00 to - co , the law by which the value of the func- 
tion is ascertained remaining the same on both sides of the point 
of discontinuity. Geometrical examples are found in many curves 
having asymptotes parallel to the axes of co-ordinates. Thus the 
equation to the hyperbola referred to its asymptotes is xy = c*, and 
when X passes through zero, 1/ suddenly changes from -co to + co . 
In such cases results obtained from the equation common to both 
portions of the curve, on the hypothesis of continuity are applicable 
to all points (however near to those where the discontinuity occurs), 
except the actual points of discontinuity themselves. After reading 
the following chapter, the student will be able to see that such 
general results may be said to have an application in ike limit to 
these points of isolated discontinuity themselves. 

There are other forms of discontinuity, but the ordinary alge- 
braical functions as ax*", sin ar, log x, are continuous except for par- 
ticular values of the variables. In all the propositions which follow, 
the continuity of the functions treated of will be assumed, except 
when the contrary is expressly stated. Our results, therefore, cannot 
be applied, without a separate investigation, to discontinuous func- 
tions except within their continuous limits. 



VARIABLES AND FUNCTIONS. $' 

5. A function of two or more variables is a quantity whose 
value depends upon and changes with the values of those variables, 
so that when arbitrary values are given to the variables, the func- 
tion receives a definite corresponding value. Thus, the distance 
which a man walks is a function of his rate of walking, and of the 
time in which he performs the distance, and the algebraical ex- 
pressions (x + i/)"',x^, sin f-j are functions of a: and y, and 

log i ^ . sin r^ 

a function of the three variables x, y, and z. 

Functions of two or more variables are represented by the sym- 
bols F(x, y)j ^ (Xf y, z), /{x, i/j Zf w), &c. Thus the equation 
z — F{x, y) signifies that s is a function of the two variables x and y. 
Those who are familiar with geometry of three dimensions, will 
recognize an illustration of such a function in the ordinate of a 
surface. 

6. In the equation F(x, y) = Oj x and y are not functions of 
two variables, but each is simply a function of the other, since on 
giving an arbitrary value to either, the other receives a definite cor- 
responding value. ^ Thus if x and y are co-ordinates of a circle, y is 
a function of x alone, whether the equation be written in the form 

y = Ja''-x' or a?' + / = fl^ 

And in general if any two variables x and y are connected in any 
manner, so that the value of either depends on that of the other (as 
is the case with the co-ordinates of a curve), this relation may be 
expressed in either of the following forms, 

(1) F{x,y) = 0, (2) y=f(x), (3) x = ct>(y), 
where the forms of the functions -F,/, </>, depend upon the nature of 
the law connecting the variables. 

When this connexion is expressed by an equation of the form 
(1), each variable is said to be given as an implicit function o£ the 
other ; in equation (2), y is said to be given as an explicit function 
of a?, and in equation (3), x as an explicit function of y. 

So also if we have an equation between three variables, of the 
form 

F(x,y,z)=^0, 



6 VARIABLES AND FUNCTIONS. 

each variable is given as an implicit function of the other two, the 
corresponding explicit functions being of the form 

and similar remarks apply to functions of any number of variables. 

It is evident that the forms of the functions^ <p, yp-, depend on 
that of F in any particular case, and in many instances the methods 
of Algebra enable us to determine them from it. This process of 
expressing one variable as an explicit function of the others vv^ith 
which it is connected by any equation, is called solving the equation 
with respect to that variable. Thus, for instance, if a:, 1/ are the 
co-ordinates of a point in an ellipse referred to the centre as 'origin, 
they are connected by the equation 
jjs -.2 

from which we derive the equivalent equations 

giving y and x respectively as explicit functions of the other. 

7. Although it follows from our definitions that a constant 
cannot be a function of a variable, yet it will be found subsequently 
that many of the propositions which will be established with respect 
to functions, will apply to constants as a particular case, in a manner 
exactly analogous to that in which the properties of symbols of mag- 
nitude are extended to zero, although the symbol of no magnitude 
at all. 

8. If in any equation between two variables, as (1), (2), or 
(3), we give to either variable an arbitrary value, the other assumes 
a definite value dependent on the former. For this reason such an 
equation is said to be an equation of one independent variable, since 
we are at liberty to give an independent or arbitrary value to one 
variable only. 

If three variables are connected by a single equation, and we 
give an arbitrary value to one of them, there exists an indefinite 
number of pairs of values of the other two which, together with the 
arbitrary value of the first, satisfy the equation ; but if a second 
variable is arbitrarily determined, the third assumes a definite value 
dependent on the arbitrary values of the other two. Such an equa- 



VARIABLES AND FUNCTIONS. 7 

tlon is therefore called an equation of two independent variables ; 
and generally, if any number (w) of variables are connected by a 
single equation, there are (w — 1) independent variables. 

If three variables are connected by two equations, as f{xy y, z) 
= 0, and <i> (x, y, 2) = 0, and an arbitrary value is given to one of 
them, the values of the other two become definite, and there is 
therefore only one independent variable ; and generally, if n varia- 
bles are connected by r equations, and we give arbitrary values to 
(n-r) of them, we obtain r equations among the remaining r varia- 
bles, which are therefore determinate. In this case, therefore, the 
number of independent variables is (m— r). 



CHAPTER II. 

LIMITING VALUES OF FUNCTIONS. 

9. Let f{x) be any function of a variable x, a any particular 
value which we may give to the variable. 

Then if by giving to a; a value sufficiently near to a, we can 
make f{x) differ from some other value J. by a quantity less than 
any assignable quantity, -^ is called the Hmiting value of /(x) 
when X approaches a, or, as it may be concisely written, 

A=:lt^f{x). 

It will easily be seen, that if /(a) is the actual value o£ f(x) 
when a: = a, this limiting value A will equal /(a) ; for/(j:) ~'/(a) 
may clearly be made as small as we please by giving to a? a value 
sufficiently near to a. Hence it follows, that in general the defini- 
tion above given of the limiting value of a function, when the 
variable approaches a particular value, is nothing else than the 
actual value of the function when the variable already equals that 
value. 

Thus, for example, let /(x) be the number of miles which a 
person, walking at the rate of 4 miles an hour, can walk in x 
hours, whence we havef^x) = ^j*. When the time is taken equal to 
2 hours suppose, the distance will evidently be 8 miles. This then 
is the actual value of f(x) when x = 2. But this is also the limiting 
value o£/(x) when x approaches 2; for the more nearly the time 
approaches 2 hours, the more nearly does the distance approach 
8 miles ; and we may, by diminishing the former difference, dimi- 
nish the latter as much as we please. Hence, by our definition, 8 
miles is the limiting value of /(a;) when x approaches 2 hours. 

The question then naturally arises — What is the use of the some- 
what complex definition of a limiting value of a function, when it is 
in no respect different from the actual value of which the idea is so 
much more obvious ? Why speak of the value that f{x) approaches 
when X approaches a, when we might so much more simply speak of 
the value which /(a;) has when x has the value a? The answer 
to this is, that there frequently are particular values of the variable 



LIMITING VALUES OF FUNCTIONS. 9 

for which we are unable to determine the actual corresponding value 
of the function, and yet can determine its corresponding limiting 
value. And it is for such values, and for no others, that the idea of 
a limiting value is resorted to. 

In the example given above, there is evidently no value of x 
for which we cannot assign the corresponding value o£ J'^x), If 
the time is given, we can at once determine the actual distance 
walked. In this and similar cases, therefore, although the idea of 
a limiting value is as admissible as in any other, we shall always 
Confine ourselves to the simpler idea of an actual v^ue. 

The following example will illustrate the necessity and use of 
the idea of a limit. 

Let /W~ • 0). 

If we give to x any value different from a, as h, we obtain 
the actual value of the above function, viz. 

/(*)=?3r=*+'' (2>- 

If, however, we give to x the value a, and endeavour to obtain 

the value of /(a), we find/(a)=-, an expression which has no 

meaning whatever, and is perfectly indeterminate ; /(x), therefore, 
has no actual value when x = a. Nor can the difficulty be obviated 
by saying, that because 

•'• / w = ^ + «* 

and therefore, when x = aj f(a) = 2a; 

for in the above process we have first divided by (x — a), and then 
put the divisor =0, a proceeding which is repugnant to the first 
principles of Algebra, and by which any fallacy might be esta- 
blished. Thus we might on the same grounds assert that, because 
■t* - a* = a? - a, when a; = cf, 
.*. x + a = l, when x = a, 

.*. 2a = 1, whatever a is, 

a manifest absurdity, arising from our having divided by a quan- 
tity which is the representative of zero. The above function, there- 
fore, has no actual value when x = a. 



10 



LIMITING VALUES OF FUNCTIONS. 



But since equation (2) is evidently true, however near b is to 
a, provided it does not actually equal it, it follows that we may 
make f{x) differ as little as we please from 2a, by making x 
sufficiently near to a. 2a is therefore the limiting value of /(a:) 
when X approaches a» 

It is so important that the reader should have a perfectly clear 
idea of a limiting value, that we shall add the following geome- 
trical illustration. 

Let PQ be any curve, PN, NQ lines drawn parallel to two 
rectangular axes ; then, if the nature of the t 

curve is known, NQ and PN will be func- 
tions one of another. 

Let then QN=(I>{PN), 

We can therefore in general determine 
the value of tan NPQ for any assigned value 
of PN; but if PN equals zero, QN or y^ 
<p {PN) also vanishes ; and tan NPQ being 

the ratio of two lines, each of which is equal to zero, has no deter- 
minate actual value. 

Since however, when (p is known, we can obtain an actual value 
of tan NPQ for any assigned value of PN different from zero, how- 
ever small, we are able to determine the value to which tan NPQ 
approaches when PN approaches zero, i. e, the limiting value of the 
tangent. 

Let jT be a point in any line 3IT which is drawn perpendi- 
cular to PN produced, so taken that 

^ = /W=o(tan iV^PQ); 

then PT is evidently the line to which the chord PQ continually 
approaches as Q approaches P, or, in other words, PT is the limit- 
ing position of the secant PQ when Q approaches P. This is 
called the tangent to the curve at the point P ; and it will be seen 
that the investigation of the inclination of tangents is one of the most 
important applications of the Differential Calculus. 




LIMITING VALUES OP FUNCTIONS. 11 

10. From the preceding explanations the truth of the following 
axioms becomes evident. 

(1) If a function f{x) has an actual determinate value when 

lt^f{x)=f{a). 

(2) If an equation is true for all values of the variable up to 
the limiting values, it is true in the limit ; that is, if the equation 
f{x) = <p{x) is true for all values of x, however near to a, 

If one of the functions, as <p(x)y has a determinate actual value 
when x=^a, this equation becomes, by axiom (1), 
lt^f{x) = <l>{a); 

and if both /(.r) and 0(j;) have actual values when a: = «, it reduces 
itself to the self-evident form 

We shall now proceed to determine the limiting values of certain 
quantities which will be required hereafter. 

11. If PQ, is a curve of continued curvature, 

arc Pg 
^ '^^^"chordPQ-^- 

A curve of continued curvature is one in which the exterior 

angle between two tangents at P and Q gradually diminishes, and 

ultimately vanishes when Q moves up to and ultimately coincides 

with P. 

T 
Let then tangents at the points P, Q, of 

such a curve intersect in T, and draw TN 

perpendicular to the chord PQ,, Then 

chord PQ=PT. cos NPT+QT, cos NQT, 

The angles NPT, NQT are always less than the exterior angle 
between the tangents ; and therefore when Q moves up to P, each 
of these angles vanishes, and each of the above cosines becomes 
equal to unity; 

. PT+QT PT . cos NPT+ QT. cos NQ T 

' ' "^^- chord PQ " "^«='> chord PQ 

= 1, by the above equation. 




12 



LIMITING VALUES OP FUNCTIONS^ 



But the chord PQ, the arc PQ, and the broken line PT + QT, 
are in order of magnitude always, and therefore in the limit; 

• •^^^*'=« chord PQ^- 
1, where x is the circular measure of the angle. 



12. IL 



sm X 



Let PQ be the arc of a circle whose centre is C ; draw CNA 
perpendicular to the chord PQ, and let the angle PCQ = 2x : whence 
we have 

PN=iPQ, AP = iaYcPQ, and ACP=x, 



Hence 




sin a; _ NP _ chord PQ 
*** a; 'IP' BXcPq ' 
J sin oc _, chord PQ, 



J tan X J sm x 
= 1, 



arc PQ 
1 



cos X 



since 



sin ar 

= 1, and cos (0) = 1. 



33. IL 



3:"-l 
x-l 



Whether n is integral or fractional, positive or negative, it may 
be expressed in the form ^ — ^ , where p, q, and r are positive 
integers ; 



y-7 
j:"-1 X ' -1 



x—1 x~ 1 

zP-9 _ 1 



if 2" 



2' Z"" - 1 



LIMITING VALUES OP FUNCTIONS. 13 

Therefore, by dividing numerator and denominator by ;s- 1, we 
obtain 



X" 



1 1 (2'^' + zP-^+ ... + 1) - (Z^-' + Z''-^ + ..♦ + 1) 



a;-l ^ ^''-^ + 2'-*+ ... + 1 

Now, when ar= 1, 2=1, and the actual value of the right-hand 
side of the above equation, and therefore the limiting value of the 

other side, is ~ — - or n, 
r 

I 

14. //^o (1 + '^)' = ^> where e is the base of the Napierian system 
of logarithms. 

Assume - = w. Then 

Also by the binomial theorem, we have 

/ IV , 1 n,n-l 1 

(l +-) =l + n. _ + ———— ._+ &c. 

\ nj n 1,2 n' 

When n becomes = oo , the part of this series which involves 
negative powers of n vanishes, and the series becomes 

= 1 + 1 + T-TT + , ^ ^ + &C. 

1.2 1.2.3 



15. .^'^^^.og.e^jA- 



For !2i^=,og.(lH..A 

X 

:=log«e by (14) 

1 

"" log a * 



14 LIMITING VALUES OF FUNCTIONS. 

16. U^'^=loga. 

Let a' -l=^z, 

and when x=0, z also vanishes, 



= log«, by (15). 

The reader will observe, that none of the above quantities have 
determinate actual values for the particular values assigned to x. 



CHAPTER III. 

DIFFERENTIATION. 

17« Let X and^ be two variables connected by the equation 

i^(^.y) = o (1), 

and let the equivalent explicit equations be 

J^=/W (2), ^ = ^(y) (3). 

Let X and y^ x + A,r and y + Ay, be any two pairs of values of 
the variables which satisfy the, above equations. The quantities Aar, 
Ay are called respectively the increments of x and y. Then, em- 
ploying equation (2), we have 

whence Ay=/(x + Ax)-'f(x) (4), 

If we suppose X] and y to remain unaltered, while Ax and Ay 
vary, equation (4) gives Ay as an explicit function of Ax. 

In the same manner we might have obtained, from equation 
(3), the equation 

Ax = (p(y+Ay)-(p(y) ...(5), 

giving Aa; as an explicit function of Ay, an equation which is 

obviously equivalent to (4). From the equations (4) or (5) we can 

Aw 
express the ratio -^ as a function of Ax or Ay. This ratio will 

generally change in value as A.r and Ay diminish, and since Aa' 
and Ay vanish together, it will ultimately assume the indetermi- 
nate form -r . Since, therefore, the above ratio ceases in this case 

to have a determinate actual value, we must have recourse to the 
idea of a limiting value, as explained in the previous Chapter. Let 
then dxy dy be any two quantities whose ratio is equal to the limit- 



1 6 DIFFERENTIATION. 

ing ratio of the increments Ax, Ay, when these increments approach 
zero, so that 

-T- = U -IT- when Aa: and Ay approach zero, 

dx and dy are called the differentials of a: and y. 
From equation (4) we have 

It -^ when Ao? and A^ approach zero = Z^a=o "! "^^ > - > • 

This limit must be a function of x dependent on the form of (/), and 
is therefore written /'(a:). Being the coefficient by which the differ- 
ential of X must be multiplied to give that of y, it is termed th^ 
differential coefficient ofy with respect to x. Hence we have 

d^ =/'W <fc, and/'(^) = It^'&l^fM . 

From equation (5) we might have obtained, in a similar manner, 

d. = <1>'Q>) dy, and ^ = It^ tkl^^lzlM , 

(p\y) being called the differential coefficient of x with respect to y, and 
being obviously the reciprocal of/' (a?). 

The object of the differential calculus is to investigate the ratio 
of the differentials, or, what is the same thing, the value of the differ- 
ential coefficients of variables connected by equations of different 
forms. From the preceding explanations the following definitions 
become intelligible. 

Def. When two variables x and y, connected by any equation, 
receive corresponding increments, any two quantities whose ratio 
equals the limiting ratio of these increments when they approach 
zero are called the differentials of the variables ; and the coefficient 
by which the differential of a? must be multiplied to give that ofy, is 
called the differential coefficient of y with respect to x ; and its reci- 
procal, by which the differential of^ must be multiplied to give that 
of a?, is called the differential coefficient of x with respect toy. 

As differentials are defined merely by their ratio to one another, 
their actual magnitude is perfectly arbitrary; this, however, does 

* 'ByltA=o is meant the limit when all the increments approach zero. In the above 
case since Ax and Ay necessarily vanish together, it is the same thing as //A,=05 or 

l(Ai,=Q. 



DIFFERENTIATION'. 



17 



nbt render an equation involving differentials indeterminate, since 
their relative magnitude is definite, and since, from the nature of the 
definition, it is impossible for a differential to appear as a factor on 
one side of an equation without another connected with it appearing 
on the other. 

18. It is sometimes convenient to view the differential coefficient 
ler the following aspect. 

If X and y were so related that A^ and Ay were always in a 
constant ratio (as is the case when x and y are the co-ordinates of 

a straight line), the ratio -—- would be the measure of the relative 

rates of increase of the variables. When, however, Aj: and Ay do 
not preserve a constant ratio, the relative rates of increase of the 
.variables will be properly measured by the limiting ratio of these in- 
crements. Hence the differentials of the variables are proportional 
to their respective rates of increase, and the differential coefficient of 
y with respect to x measures the relative rates of increase of y 
and X. 

19. As has been already observed, every equation between two 
variables may be regarded as the equation to a curve, and every 
proposition founded on such an equation admits of a geometrical in- 
terpretation. The following geometrical illustration will aid the 
reader's conception of the definitions of Art. 1 7. 

Let PQR be the curve corres- 
ponding to the equations (l) (2) 
and (3) of Art. 17. 

Since x, y and x + Ao?, y 4 A^ 
are taken to be values of the va- 
riables which satisfy the equations, 
they must be the co-ordinates of 
two points in the curve. Let these 
be P and Q so that we have 

OM = x, ON = x + Ax, MN=Ax, 
Om = MP==y, On = NQ = y+Ay, mn = QU=Ay. 

Let Pr be the tangent at P, T any point in it. 

Now suppose Q to move up towards P. As it does so, the ratio 
Ay QU . , TF 

Ax °^ RU "^^^"^^^ ^^^^ ^^^ "^ore nearly equal to -p the trigono- 
metrical tangent of the angle which PT makes with the axis of x. 
H. D. c. 2 



71 



K. 



M 



N 



1 8 DIFFERENTIATION. 

When Q reaches P, QU and PU both vanish, and the ratio -p^ 

ceases to have any actual value^ after having approached indefinitely 

TV 
near to pj^ . Hence 

TV and PV therefore satisfy the definitions of dy and dx. 

Since this is equally true at whatever point of the tangent T is 

taken, TV and PV may have any magnitude we choose to assign to 

them, the only restriction imposed by the figure being that they shall 

be lines parallel to the axes and bounded by a point in the tangent 

to the curve. 

The differentials therefore are not indefinitely small (as they are 

sometimes represented) but of a purely arbitrary magnitude, their 

relative magnitude only being fixed. 

Their ratios, that is, the respective differential coeflScients are 

evidently given by the equations, 

/' W = ^= tan rPr= cot TPv, 
PV 

that is, f'{x), (p\y) are the tangents of the angles which the tangent 
of the curve makes with the axes of x and y respectively. 

The geometrical meaning of Art. 18, is equally clear. The 
assertion that dx and dy are proportional to the respective rates of 
increase of x and y merely amounts to this, that TFand PV measure 
those rates of increase at the point P, the increase of y being pro- 
portionally greater as the inclination of the tangent to the axis of x, 
increases, and vice versa. The truth of this is apparent from the 
figure. 

In the case we have taken, x and y are increasing together at the 
point P, and all the quantities 

A^ {PU) Ay (QC7) dx (PV) and dy {TV), 
and the tangent of the inclination of P J* to 
the axis of x are positive. If an increase of 
X had corresponded to a decrease of y^ the 
figure would have been as below in which dx 
and dy have opposite signs, and the tangent 
of the inclination of PT to the axis oi x is negative. 




\\ 



DIFFERENTIATION. 1 9 

Hence, f'{x) is positive or negative according as an increase of a; 
produces an increase or decrease oi y. 

20. The remainder of this Chapter will be devoted to the deter- 
mination of the ratio of the differentials of variables connected by 
equations which can be reduced to an explicit form. The treatment 
of equations in an implicit form will be deferred to a subsequent 
Chapter. 

It may be remarked here, that the process of finding the differ- 
ential or differential coefficient of any function is called differentiating 
the function. 

21. Let y=w, +^2+ -^u^i 

Ui, u^ M„ being all functions of x. 

Let X receive an increment Aar, and let the corresponding incre- 
ments oiyy Wi Wn, be Ay, At^j Az<„, Then 

^ + A^ = Wi + AM1 + M2 + AMg +2/„ + Aw„, 

.-. Aj^=Ami + Am2+ A?/„; 

Ay _ A?^i Awj Ae/„ ^ 

" A^~ A^"*" A^ ^~Ex' 

which, being always true, is true in the limit. 

^ dy dui dua du,. 

' dx dx dx " ' dx' 

or dy = dui + du^ + . . . du^ ; 
that is, the differential and differential coefficient of the sum of a 
number of functions are equal to the sum of the differentials and 
differential coefficients respectively of the several functions. 

22. Let y -f(x) + C, where C is independent of x and y. 
Then, if Ay, Af(^x) be the increments of y and J^ix), corresponding 
to an increment Ax of x, 

A^ = A/(x), 

since the change of x into x + Ax does not alter the value of C, 

. Ay ^ A/(^) _ 
* * Aa; Ao: ' 
therefore, in the limit, 

dy df(x) 
dx dx ^ 

or dy^dfix). 

2—2 



20 DIFFERENTIATION. 

Since we should have had by (21), if C had been a function of a', 

dy = df{x) + dC, 
this result may be expressed by saying that the differential of a con- 
stant is zero. 

23. Let ^ = Wi . u^, 

Ui and Ug being functions of x. Then, with the same notation as 
before, 

i/ + Ay = (u, + Am,) (w2 + Amjj), 

Aw Awo Am, Am, ^ 

When we approach the limit by making the increments approach 
zero, the last term becomes -—.It^AUs', which (since -5- is gene- 
rally finite, and ltAu2= 0) equals zero, 

dy _ dug dui 
' ' dx ^ dx ^ dx ' 

or dy = M, dus + u^ du^ . 

24. This rule may be extended to the case where y is the pro- 
duct of any number of functions of x. Thus, let 

«/ = Wi.M2...W„_l.t^„. 

Then, by (23), dy = du^ . {u^ . . .u^ + u^d{u^... m„}, 
^ </{mi...m„} ^c?m, ^ d{u^...u^} 

Ui. . .U„ III Ug. . .u^ ' 

^ d{u2...u„} ^du^ _^ d{u3...u„] 

U2..,U„ U2 M3.. .u„ 

By substituting this value in the former equation, and continually 
repeating the process, we obtain 

rf {m, . . . M„} _ du^ du2 du„ 

a theorem of which the result of the preceding article is a par- 
ticular case. 

25. If y = Cf{x\ where C is independent of x and y, 

m 

Ay = CAf{x), 



DIFFERENTIATION. 21 

• • A^ ~ Ax ' 
which being true in the limit, we have 

dx dx ' 

or dy = Cd/(x) ; 

that is, if a function of x is multiplied by a constant, its differential 
and differential coefficient are multiplied by the same constant. 



26. 


Let 1/ 


W2' 




With the usual notation, we shall have 






^ + Aj/ = 


w, + Aui 
' Us+Au/ 






... Ay = 


Ui + Am, Ui 

' Us + Au^ Us * 
UsAui - u^Au.2 








Usius-vAu^ * 






\ ^y 


Aui Aus 






" Ax~- 


Us (^2 + AWg) 


Therefore, 


in the limit 


, we have 






dy ^ 
dx~' 


dui dur. 






^^^ u^dui — Uidui 

^ ~ Us' 



27. Let y = x^, n being any number whatever. 

Then Aj/ = (a? + Aa;)" - j;", 

Ay _ (a; + Ax)" - j;" 
' ' A^~ Ax 



fi + ^Y-i 



22 DIFFERENTIATION. 



Aic 
Let 1 + — =z. Then when Ax approaches 0, z approaches I ; 

•• dx-'^ ^^^' z-1 * 
The value of this limit has been found in (13), and =w, 

dx ' 

or dy = n «"~^ dx. 
From the results of this and former articles, we have, if 

y = a ->rbx + cx^ +px^) 

dy =^{b -{- 9>cx + + npjf~^} dx» 

28. Let y = log^ x. 
Then Ay = log^ (^ + Ax) — log« x ; 

. X + Ax 

, Ay ""^v X 

"Ax Ax 

^1 log.{l+4 if ^^2. 

And when Ar approaches zero, z approaches zero; 
" dx'x^^'-' z 



This limit has been found in (15), and =^ ; 

^ ^ log a 

dx a: log a' 
, 1 dx 

or ay =:; . 

•^ log a X 
Hence, if ^ = log x, this becomes 
, dx 

log X meaning the Napierian logarithm of x. 



29. 


Let y = a*. 




Then 




Ay = a'+^*-a'; 
Ay_ ^ a^^-1 
" Ax~^ ' Ax 






DIFFERENTIATION. 23 

The value of this limit has been found in (l6), and = log a. 







••• !='<'«''•«', 


or 




dy = log a . a'^dx. 


Hence if 




y = ^, 
dy = e*rfjr. 


30. Let y 


= sin X* 




Then 


A5,= 


: sin {x + A.r) - sin x 




= 


■- 2 cos {x + i Ajc) sin | A:r ; 






, , ^ sin H Ax 
= cos(^+^Aa:). ,^^ , 




-. %- 


, , , ^ ^ sin s Ax 
: /<A=o COS (x+^Ax) -r--— 



sin j^ /\ .y 
and //^^ COS (or + ^Ax) = cos j:, and /^a=o "Ta — = ^» ^^ (^^)' 

.*. -r = cos a?, 
ax 

or rf^ = cos a; dx, 

31. Let ^ = cos X, 
Then A^ =^ cos (^ + Ar) - cos a: 

= - 2 sin (a: + J Ax) sin | Ax ; 
Ay , , . . . sin H Aa? 



^y 1, • / 1 A N sin iA« 



2 

sin^Aj: 



and //a=o sin (a? + ^ Ao;) = sin x, and //a_o — ~^ = 1* by (1 2) 

o i\x 



dy 

or dy — — sin areir, 

32. Let y - sec or. 
Then Ay = sec (a; + Ax) — sec a: 

1 1_ 

cos {x + Ax) cos X * 

2 sin {x 4- JAa?) sin ^Ax ^ 
cos 07 . cos {x + Ax) ' 



24 DIFFERENTIATION. 

Ay sin(x + ^Aa:) sin f Aj? 

Aar cos x . cos {x + Ax) ' ^ Ax 

, , sin(x + iAx) sin X , . siniAa: . . . 



rfj; 


sin X 
cos^ x 


> 


d5,= 


sin X 
cos^x 


rfx. 



or 

33. Let y = cosec x. 
Then A^ = cosec (x + Ax) - cosec x 



sin (x + Ax) sin X 

2 cos (x 4- ^Ax) sin | Ax 
" sin X . sin (x + Ax) ' 

Ay cos (x + 4 Ax) sin | Ax 

Ax sin X . sin (x + Ax) ' 5 Ax ' 

, - cos (r + 1 Ax) cos X , ,^ sin ^ Ar , , ,^. 

and /<^_o -. \ ,^ /. = -^^— , and li^^ -^ = 1, by (12), 

=° sinxsin(x + Zix) sin^ x ^^ ^Ax J\y> 

dy cos X 

dx sin* X ' 

, cos X _ 

dy = r-5— . dx, 

^ sin^x 

34. Let 3/ = tan x. 

Then A^ = tan (x + Ax) - tan x 

_ sin (x + Ax) sin x 
cos (x + Ax) cos X 

sin Ax 
cos X .cos (x+ Ax) ' 

A^ _ 1 sin Ax 

Ax cos X . cos (x + Ax) ' Ax ' 

""■* '"^^ cos (A A..) = E^' ""•* ''^=«^^=i' •'y (>2)' 

.'. -f- - sec^ X, 
ax 

or f?y = sec'* X </x. 



DIFFERENTIATION. 25 



35. Let y = cot x. 

Then Ay = cot (x + Ax) - cot a? 

cos (a; + Ax) cos a; 
~ sin (x + Ax) sin x 

sin Ao: 



and // 



sin X . sin (a: + Aa?) ' 

Ay 1 sin Ao; 

Ax sin a? sin (x + Ax) Ax ' 

1 1 1 7. sin A^ , , ,,„>. 



~° sin {x + Ax) 

.', -J- = — cosec X, 
ax 

or dy^ — cosec* a? </a:. 

36. Let y — sin~' x. 

Then j: = sin y, 

.*. </a; = cos y dy^ by (30), 

.'. dy = dx; 

^ cos y 

or, expressing the differential coefficient in terms of x, 

37. If it is required to find the differential of the inverse sine 

without assuming that of the direct sine, we must proceed as 

follows: 

y = sin~^a?, 

.*. smy = Xi 

.*. sin (jy + Ay) — siny = Ax, 

.*. 2 cos {y + 5 Ay) sin J Ay = Ax, 

, ... sin AAy Ax 
... cos (y + 4 Ay) -^^^^ = ^; 

and /<A=o cos (y + i Ay) = cos y, and /<a=o ^^"^ ^= 1, by (12), 
... ^-cosy=^(l-^''). 



26 DIFFERENTIATION. 

Since the steps in the determination of the differential of the 
inverse function are precisely the same as for the direct function, it 
is needless to repeat the independent proof for cos~* j?, sec"^a?, &c., 
as the reader will be able to supply it without diflSculty when re- 
quired. We shall therefore deduce the differentials of these func- 
tions from those already found. 

38. Let y — cos~^a7. 

Then x = cos y, 

.*. dx— — %\x\.ydy^ by (31), 

.*. dy = — i — -dxy 

^ sm y 



or 



39. Let 5/ = sec-'^. 




Then x = sec y, 




sm y , 


by (32), 


... dy = ''''''^dx- 

^ smy 


1 


tan y sec y 


dy= — 77-0 — - 


,dx. 



dxj 



xJ{o^-\) 

40. Let y = cosec"^ x. 
Then x = cosec y, 

••• ^•^ = -^^^^' by (33), 

- sin^y , 1 , 

,\ du = — dx = dx, 

^ cos y cot y cosec y 

or dy = TT-j — T^dx. 

41. Let ^ = tan~^ a?. 

Then x = tan ^, 

,•. dx = sec'^ dy, by (34), 

.% dy = — 5— dx, 
^ sec' y 



or dy = 5 dx. 

if 1 + 



1 



DIFFERENTIATION. 



27 



42. Let y = cof' x. 




Then a? = cot 3^, 




.*. dx = - cosec^ y dy, 


by (35), 


•■• ''^ cosec^^''^' 




d!> ^^^,dx. 





or 

43. Let y = versin"^ x. 

Then a; = versin y = l- cos y, 

.*. dx = smy dy, by (31), 

.•. dy = -7 — dxj 

^ sm y 

and sin^ = 7(l-cos^y)=V{l-(l-^)'} = x/(2^-^). 

44. Let y = suversin"^ a:. 

Then x — su versin y — l+ cos y, 

.', dx= — sin ydy J by (28), 

),\ dy = — ; dx, 

^ %m.y 

and sin y=^Ji\- cos'y) = J{l-{a;- 1)'} = J{9,x - x% 

45. We have now shewn how to differentiate all the simple 
functions which ordinarily occur in analysis, as well as any func- 
tion formed by the addition or multiplication of any number of 
them. It remains to differentiate functions compounded of these 
simple functions, such as log sin x, d' ^"^ *^% cos (« + hxy, &c. All such 
functions are included in the general forms /{^ (a;)}, /|30{\|/'(j;)}], 
&c., where/, ^, -v/^, &c. denote simple functions already differentiated. 
The following theorem will enable us to differentiate such compound 
functions. 

46. Let y=f{<p(.^)}' 

Denote ^ (ar) by w, so that the above equation gives 
u = 4>{x\ 
y=f(u). 



28 DIFFEREXTIATIOX. 

Let Ax, Am, Ay be corresponding increments of ar, Mj y> all of 
which evidently vanish together. 

If du is the differential of u corresponding to the differential dx 
of X, it is determined by the equation 

du = 0' (x) dx ; 
and if rfy is the differential of y corresponding to du, it is determined 
by the equation 

dy=f{u)du. 

And, substituting the above value of du, we have 

dy=f{u)<p'{x)dx, 

which gives the required relation between dy and dXff{u) and ^'(x) 
being determinable by our previous methods. 

47. The correctness of the above result may not appear quite 
obvious, because we have in effect defined dy, not by the funda- 
mental definition 

1="-^ (« 

but by compounding the two definitions 

-S=''-^. («) 

-^l=^'-S w 

This is however immaterial since the last three equations are not 
independent, any one of them being deducible from the other two. 
Thus from (2) and (S) we can obtain the same value of dy as that 
given by (1). 

For % = Ay ^. 

Ax Au Ax' 

•• "^='»Ai-''^='»A^^'^=*Ai- 

By equations (2) and (3) this becomes 

// ^_^ dM_dy^ 
""^'Ax'du dx'dx' 

which is identical with equation (1). 



DIFFERENTIATION. 29 

48. The following proof of the above proposition is free from 
the apparent diflSculty. As before, let 

« = </>W. (1-) 

and .: y=f{u) (2) 

Let Ax, Am, A^ be corresponding increments of x, u, y, which 
all vanish together. 

Then ^ = '^-^=°a:^ = ^^'^-A« Ai- 



But from (l) and (2), 

dy 



^^-o^u^f{u), and //^=o|^ = 0'(x); 



and dy=f{u)(p'{x)dx. 

49. The last article gives the proposition in a form which, consi- 
dered as a mere proof of the theorem, is preferable to that of Art. 46. 
The former proof has been retained for the purpose of shewing more 
clearly the connexion between the equations (1), (2), (3) of Art. 46. 

The student will perceive that such equations as 

dy _ dy du 
dx du dx"* 

may be used without fear of error. 

This is self-evident, provided the differentials which enter twice 
have the same value in both places, which (as appears by Art. 4?) 
will be the case, even though they are defined on the one side by 
equations (2) and (3), and on the other by equation (1). 

50. The introduction of the new symbol {ii) for (p (x) may be 
dispensed with, when the process of differentiation will take the 
following shape, 

'.dy = df{<p{x% 

= f{<p{x)]d<p{x) 

/' {0 W} being the same as /' («), that is, the same function of <p (x) 
which /' (x) is of x, and therefore determinable by the rules above 
investigated. 



30 DIFFERENTIATION. 

51. In like manner we may differentiate z, where 

For putting w = (x), 

and y=/(w), 
we have z = F{y); 
,'.dz = r(y), 
dy=f{u)du, 
du = 0' (x) dx ; 
.-. dz-^¥{y)f{u)<^\x)dx,^ 
or without introducing any new symbols, 

dz = r [/{0 (a.)}]/ {0 W} <^' {x) dx, 

and the theorem may evidently be extended to expressions com- 
pounded of any number of functions. 

52. The following examples may serve to illustrate the method 
of differentiating any explicit function of one variable. 

(1) Let y =/ {x) = ha' + car'. 

Then dy = bda" + cdar^, by (25), 

and da" = log a . d'dx, by (29), 

and da~^ = c?a", if u = — x 
= log a.(t du 
= '[og a.a~''(—dx) 
.'. dy = log a {^a" — ca~"} dxy 
and /' (j;) = log a {6a^ ~ ca~"}. 
The actual introduction of a new symbol is useless where the 
function of x, which it represents, is not very complicated. Thus 

(2) Let y = log sin x -f {x). 
Then dy = d log sin x 

,\ dy = ■ — -. = cot X dx, by (30), 

and /'(j;) = cotx; 

where the process is evidently the same as if we had substituted 

u for sin x. 



I , by (29) and (46), 



DIFFERENTIATION. SI 

(3) Let^ = a^'^"8')'=/(a:). 

Then dy = log fl . a('^°8*5" d \{x log xy\ by (29), 

= log a . flC'-s')" {w (or log xf-^ d {x log x)\ by (27 j, 

= n log a . a('i°s')" oiT-^ (log x)""^ {^a? log ar + x fi log x\ by (23), 

« w log a . o(*i°8'>" a?«-^ (log a:)"-' (1 + log x) dx, (by (28), 

and/(a;) = w log a . a^*i°s'5V-' (log a:)"-' (1+ logo;). 

(4) Let ^ = tan (cot"' a*), whence ^ = - . 

We will find dy from each form of the expression as an illus- 
tration of Art. 46. 

From the second form of the equation we obtain immediately, 

since 

y = ar\ 

^^ 

dy = — x~^ dx = — J- . 

Also from the other expression we have 

y — tan w, if w = cot"' x, 

— dx 



,*. dy = sec* u du, and du = , 

X "T X 



, sec'w J 



±dx 



dx 



1+x^ 
= r , as before. 

For examples on this and other parts of the subject, the reader is 
referred to Gregory's Examples in the Differential Calculus, 

53. The results of this Chapter are here collected, and should be 
carefully remembered. 

dy = df,{x) + dflx) 4- ... dfXx\ ovf{x) =f:{x}+f,\x) + ...//(ar). 



S2 DIFFERENTIATION. 

If3^=/(a:)=/.(a.)/,W, 
dt/ =/M df,{x) +f,{x) dflx\ orf{cc) =Mx)A'(x) +/.(^)//(x). 



lf^=/W 



'fM ' 



^^^ xm? ' -^^ ^" '{/MY ' ' 

If ^ =/(^) ^fM + C, dy = dflx\ f\x) =//(.:), 

= C/,(^), dy = Cdflx), f{x) = Cf,'{x), 

= a:-, <Zj^ =:= nx"*-^ dx, f (x) = nx''-\ 

dx 1 

= loff„ a:, dy = . fM = i j 

^" ' ^ \oga,x ^ V y loga.a; 

= a", dy — log a.a' dx, f {x) = log a.a', 

= sin or, dy = cos a; ^o?, /' (x) = cos a', 

= cos £C, dy = — sin x dx, /' W = "" ^^^ "^^ 

- sin J? - ^, , . sin a; 

J cos a? , .,, . cos a: 

= cosecx, dy = ^-^— dx, fM = — ^-i — > 

^ sin^ X '' ^ '' sin* X 

= tan 0?, £?y = sec* x dx, /X^) — ^^^^ ^i 

— cot x^ dy = - cosec^ x dx, f {x) — — cosec' x, 

= cos-'*, dy= ~ , /'(x) 



= cosec-'x, (f^= ~ f_^, /'(■»•) = 



dx 1 



DIFFERENTIATION'. 33 

y=/(x) = i^(«) Bud u = ip{x),di/^F\u)(pXoo)dx, f\x) = F'{u)<p\x), 

These results are expressed both in the notation of differentials and 
of differential coefficients, in order to familiarise the reader with 
both. Both notations express the same fact, namely, that the ratio 

^ is equal to a certain function in each case, and, so far as we have 

carried the subject at present, may be used with equal convenience. 



H. D. c. 



CHAPTER IV. 

INTEGRATION. 

54. In the preceding Chapter we have shewn how to differen- 
tiate any explicit function of a single variable, that is, having any 
equation of the form ?/ =/(j'), we have shewn how to determine 
the value o? f{x) in the equation dy =f{x) dx. 

Integration is the converse of differentiation, that is. It consists 
in finding from any differential equation dy =f{x) dx the integral 
equation y =f{x), from which it has been derived. 

The symbol (/) by which this operation is represented is the 
converse of the symbol {d) which represents differentiation, and is 
therefore defined by the equation 



/' 



/- 



dy=^y. 

Hence, \i f'{x)dx\% the differential of /(a:), 

' f\x)dx=f{x), 

fix) is called the integral of /'(.r) dx^ or sometimes, though not very 
correctly, the integral off'{x). 

55. From the nature of the rules which define a differential, 
the form of /'(oj) can always be determined where that of /(x) is 

given. We have only to find {^ , and determine its limiting 

value, and fXx) is known. No such general method of integration 
can be found : for since the integral of any differential is defined to 
be the function from which it may be obtained by differentiation, we 
can integrate those functions only to which the differentiation of 
other functions has chanced to lead us. Thus, if we are required to 
integrate any function (p(x)dx, we cannot be sure that the process is 
even possible, that is, we cannot saylhat there is any function whose 
differential is (p {x) dx, unless we have observed that this quantity, or 
some other to which it is equivalent, has been obtained by the dif- 
ferentiation of some known function. It will, however, appear here- 
after that all functions of the form (p (x) dx can be integrated in the 
form of infinite series, although there are very many whose integrals 
cannot be expressed in finite terms by any of the ordinary symbols 



INTEGRATION. 35 

of analysis. The method to be pursued in integration will therefore 
be, first to collect a number of integrals by examining the results of 
the differentiation of some simple functions, and then by various 
artifices to make the integrals of other functions depend upon those 
so determined by inspection. In this way large classes of functions 
have been integrated in finite terms, although many are incapable of 
such reduction. 

56. Since (Art. 9.2) f{x) + C and f{x) have the same differential 
f{x)dx, it follows that we may take as the integral o? f {x) dx 
either of the above quantities, giving to C any value we please in- 
dependent of X. Hence it appears that while there is only a single 
differential of a given function, there is an indefinite number of in- 
tegrals of a given differential, all of which are included in the general 
form/(x) + C, where C may have any value independent of a? which 
we choose to assign to it ; C is called an arbitrary constant, and must 
be added to every integral in order to express it in its most general 
form. Where, however, the form of (/) is the only object of the in- 
vestigation, the particular integral f{x) is often spoken of intead of 
the general form /(x) + C. 

57. By considering the differentials of f(x) and x as measures 
of the rates of increase off(x) and x respectively (as explained in 
Art. 1 8), the problem of differentiation may be said to be — Having 
given a function of x, to compare its rale of increase with that of x; 
and that of integration, Having given the ratio of the rates of in- 
crease of the function and the variable, to determine the function. 

Now it is evident that, for a given function, the rates of increase 
can have but one definite ratio for a given value of the variable, 
whereas an indefinite number of functions, differing only by constant 
quantities, will have the same value of this ratio. 

Thus we may draw as many parallel straight lines as we please, 
in all of which the rates of increase of the co-ordinates have the 
same ratio, since they depend only on the inclination of the line and 
not at all upon its distance from the origin. If y = mx is the equa- 
tion to one of these lines, they are all included in the general form 
y = mx^rC, where C is perfectly arbitrary, and all have the same 
value of the differential coefficient. The equation to each may there- 
fore be obtained by integrating the same differential equation 

dy — mdx. 

3—2 



3 6 INTEGRATION. 

To take a more general case. Referring to the figure of Art. 19-, it 
is clear that an infinite number of curves may be drawn having 
their tangents at the extremities of a common ordinate, equally in- 
clined to the axis of x. The equation which includes the whole class 
of curves is i/ =f{^) + O, where C is arbitrary, since the addition of 
a constant to every ordinate merely changes the distance of the curve 
from the axis of Xj without altering its form or the inclination of its 
tangent. In all these the differentials therefore have the same ratio 
for the same value of jf, and any one of the equations may be ob- 
tained by integrating the differential equation dt/ =f'(^x) dx which is 
common to them all. These observations shew the meaning of the 
proposition established in the last article. 

We shall now proceed in the empirical manner indicated in 
Art. 55. f to determine the integrals of the differentials given in 
Art. 53., and some others immediately deducible from them. 

58. Since 

^{/iW +/2W ... +Mx)} = df,{x)^dMx) + ... d^x), 

••• /i W +/.(^) +/. W = \WM + d^x) + ... df,Xx)}, 

or |{//W +//(.:) + . . .fj{x)} dx = jf/{x) dx 4 jf/(x) dx...+ l/,:{x) dx, 

that is, the integral of the sum of any number of functions equals 
the sum of the integrals of the several functions. 

59. Since d C/{x) = C df(x), 

.'. C/{x)=jCd/(x)=jCf(:v)dx, 

or fcf(x) dx=C ff(x) dx ; 

that is, if the quantity to be integrated is multiplied by a constant 
factor, the constant may be placed outside of the sign of integration. 

60. Since d (x") = nx""'^ dx, where n is any number, integral or 
fractional, positive or negative, different from zero. 



''-'dx = -+C, 



x"dx= — - + e, 

n+ 1 



where w is any number different from - 1 ; that is, to integrate 
any power of x except - we must increase the index by unity and 



INTEGRATION. o7 

divide by the index so increased, an arbitrary constant being added 
to obtain the general integral. 

61. Since d}oo'x = — , 

X 



I 



— = log a: + const. = log C-r, 



I' 



if we write the arbitrary constant in the form log C. This deter- 
mines the integral of x" in the case excepted from the previous 
article. Hence 

X x^ x'"- 
= a + b — . . . + ex + d\og X - - . . . - ~. f> ^_. + C. 

62. From j — we can find I ,, „ ^ as follows. 

Let x^ ± a^ = z<^ 

.*. xdx = udu, 

dx du dx + du 
' ' u X X + u ' 



f dx fdx fd (x + ii) 

'** J J{^'^^')^ J~u^ j x + u ' 



= log (x + u) + const., 

° a 

C . . 

writing the constant in the form log — to give the integral a more 

convenient form. 

63. From the same integral we can also deduce the values of 
f dx ^ f dx 

a^ — x^ 2a {a + X a — x) 

and d {a~x)= — dx, and d(a + x) = dx; 
f dx 1 f dx 1 f dx 

' ' J a^-x^ ~2a J a + x 2d J a-x ' 

1 fd(a + x) 1 fd(a-x) 
2a J a + x 2a J a~x 



I-. 



2a 

= Ya ^""^ (« + *^) - Q- ^^§ {ci-x) + const. 
dx 1 , ^a + X 



88 INTEGRATION. 

f dx _ I C dx 1 r dx 
" J x^-a^~ 9.a J x~a 2a J x + a' 

_ 1 Cd{x-a) 1 [d(x + a) 
~ 9.a J x-a 2a J x + a * 

— 5" ^^^ (-^ ~ ^) ~ ^ ^^S (-^ + «) + const. ; 
^a /ia 

f dx 1 , ^x — a 

/dx 
77-^ FT- 
xj(a'^x'}^ 



For let j: = -, .*. log x = log a — log u ; 



dx du 

X u ' 



J{a'^a^) 



•A"ff 



•*• jxjia'^x') y 



du 



1 /• du 



therefore^ by Art. 62, 

. f___^ 1 1 ^ ^ 

65. Recurring to Art. 53, we find 

J(a'')=loga.a''rfj', 

.-. (a^dx = :i-^.a' + C, 
J log a 

If rt = e, this becomes, / e'^/j: = 6^+0. 



INTEGRATION. 



39 



Again, dsinx= cos x dx^ 

cos a; dx = &mx + €• 



I' 



And since d sin mx = cos mx d (mx) = 7« cos mx dx, 



I 



cos mx dx = — sin mx + C 
m 



Similarly, d cos mx =- m sin mx dx, 



I 



sin mx dx = — cos mx + C. 
m 



Also, d tan mx = m sec^ mx dx ; 



I 



sec^ mx dx = — tan mx + (7. 

7W 



And d cot wo; = — ?w cosec^ mx dx 



/ 



cosec^ mx dx = cot mx + C. 

m 



Again, d sin M - = ,, sr = -TTi — r\ 



/ 



^0? . 1 ^* ^ 

-77-^ 5^ ■■= sin ' - + C. 

J(a^ - x^) a 



o 1 . 1 -\X \a) dx 

bo also, since d cos ' - 



C dx 1 ^ ^ 

•*• Ht-2 ix = - COS-' - + C. 

Here we have obtained a second form of the same integral, but 
since 

. ,X , X TT 

sin~' - + cos - = - = const. 
a a 2 

the two expressions include precisely the same system of values 
when all possible constant values are given to the arbitrary con- 
stants. 

A . , ,x \a/ a dx 

Again, rfsec" - =• 






dx 1 yX ^ 

-jT = - sec" - + U. 



X J(x^ — d^) a a 



40 INTEGRATION. 

As in the preceding case, this integral might have been expressed in 

the form 

1 i^ j^ 

— cosec"' - + (7, 
a a 

which may be shewn as before to be identical with that already 
given. 

Smce a tan ^ 



f dx 1 ^ _i J? ^ 

••• / -2 2 = - tan - + C. 

J or -\- X a a 



As before, we might have obtained the equivalent form 



1 ^ 1^ ^ 
- cot-' - + C. 
a a 



c,. 1 . ,x \a/ dx 

Smce a versm"^ - = 



© 



//« — i\ = versin-' - 
^{2ax - X) a 



/dx ' \^ r^ 



or = — suversin-' - + (7. 
a 



^^. Collecting the results of the preceding Articles, we have 

xx^'dx = 7+C unless « = -!, and then 

; w + 1 ' 

I ~^-; — 2 = - tan ' - + (7, 
J a^ + x^ a a 

f dx 1 , ^ « + or 



,.r-« 



ar + a 



f dx . ,x ^ ,x ^ 



INTEGRATION. 41 



dx 1 _i^ . ^ 



J X J{x^ - a") a ^^^ a 



dx . ,x 

= versm" 



J{Slax-x^) a 

a'dx = J + C, 

log a 

sin wx f/.r = cos mx + C, 

772 

cos /war f/o: = — sin mx + (7, 
m 

sec^ TWO? e?.r = — tan mx + (7, 
m 



cosec^ mxdx = cot mx + (7. 

7?Z 



The above are termed Jjcndamental integrals, and should be care- 
fully remembered. 

All other integrals are obtained by reducing them to some one 
of the above forms. 

It may be observed that where the function under the integral 
sign is homogeneous in x and a, the integral is also homogeneous 
and one dimension^ higher in terms of x and « ; or of the same 
dimensions, if we consider dx as of the same dimensions as x. 

The remainder of this Chapter will be occupied with various 
methods of reducing the integrals of several classes of differentials 
to some of the fundamental forms. 

Integration hy Algebraical Transformations. 
67. An integral may often be reduced to the form of one of 
the fundamental integrals, or to the sum of two or more by simple 
algebraical transformations. 

As an example, we may take the following : 
[ dx f dx 

ji + .v + x'~J^+{i + xy' 
^ f _i(i±f)_ 

2 ^ ,2^+1 ^ 



42 INTEGRATION. 

The student will find numerous examples of the application of 
this method in Gregory's Examples^ p. 246, et seq., with which he 
should make himself familiar. 

Integration hy Parts. 

68. The following theorem is often of great use in reducing 
integrals to the required forms. 

Since d{uv) — udv + vdu, 

u and V being any functions of a variable Xj 

.'. uv= ludv+ ivduy 

.'. tudv = uv— Ivdu (1). 

This is called the formula of integration by parts, and enables 
us to integrate any function udv if the function vdu is in an inte- 
grable shape. 

69. As an example of its application, we may take 
'x log xdx. ' 



I- 



Assume log x = u, and xdx = dv, 

dx x^ 

whence — = du, and — = v, 

X ^ 2 

the arbitrary constant being omitted in the value of v for the sake 
of simplicity, since the formula holds when v is any quantity whose 
differential is dv. The constant would, in fact, disappear from the 
result if it had been introduced. 

Hence I x log xdx= I udv, 

= uv- jvdu by (1), 

x^ , fx^ dx 

= -.log.-j_._. 



x\ x' ^ 

= jlogx-- + C, 

x^ 
= 2- {log^-il + C^. 



The student is again referred for examples to Gregory's Exam^ 
pies, which he is recommended carefully to study in all parts of 
the subject. 



INTEGRATION. 43 

Rational Fractions*. 

70. A rational fraction is a fraction of the form 

a:" + 5ia;"-^ ... +qn ' 
where the indices of x are all positive integers and the coefficients 
constant. 

If the numerator contains powers of x as high or higher than any 
in the denominator, the fraction can always be reduced by division 
to the sum of a rational integral function of x, and a rational frac- 
tion, in which the largest index which occurs in the numerator is 
less by unity than the largest in the denominator : the first part of 
this can be integrated by inspection, and therefore we need only 
consider those rational fractions in which the dimensions of the 
numerator are less than those of the denominator. 

Rational fractions are integrated by resolving them into the 
sum of a number of fractions with simpler denominators, called 
partial fractions. 

71. This can always be done as follows. Let the rational 
fraction be 

where m is not greater than w - 1. 

Let the equation V=0 have one real root equal to a, r real roots 
equal to 5, one pair of impossible roots equal to a ± /3 J(- 1 ), and s 
pairs of impossible roots equal to a'±/3'^(-l), which comprise all 
the forms in which the roots of any rational equation can occur. 

. U A B, Br., B, M+Nx 

Assume -f> = + -; rrr + 7 — y^-f . . . + V + 



F x-a {x-by {x-by-'"' x-b a^-2ax + a^+/3'' 

K, + L,x Ki + Lix 

'^ijf'-2a'x + a''+/3'j'^'*''^x^-2a'x + a''+(3'' ^^^ 

Then real values of A, Br, Br-i- . .B,, M, N, K,. ..K,, L,. . .i„ 

can always be found to satisfy this equation. For by multiplying 

both sides of the equation by V, we obtain an equation 

U-f{x) = ...(2), 

* Those readers who are not familiar with the Theory of Equations are recom- 
mended to omit the remainder of this chapter. 



44 INTEGRATION. 

where U is a rational integral function of j: of not more than (» — 1) 
dimensions, and /(a:) a rational integral function of (w — 1) dimen- 
sionSj in which the coefficients of the several powers of a: are linear 
functions of the indeterminate quantities A, Br, &c. in equation (1). 
In order that the two sides of equation (l) may be identical, the 
coefficients of the several powers of x in equation (2) must vanish. 
This condition will give us n linear equations to determine A, Bry 
&c Now the number of these coefficients is evidently equal to the 
number of roots of V=0, that is, to w, and therefore the n linear- 
equations will give real values of them which satisfy equation (l). 

Hence w may always be resolved into partial fractions of the forms 

assumed in equation (l). 

If the dimensions of U had been greater than n — 1, this method 
would have failed, because the coefficients of the higher powers of x 
would have contained none of the indeterminate quantities, and 
could not therefore have been made to vanish by assigning particular 
values to these indeterminate coefficients. 

If the equation V=0 contains more than one of each of the four 
classes of roots, w« must add partial fractions of the forms above 
given for each of these roots, and it will appear, as above, that real 
values may be found for the indeterminate coefficients which will 

satisfy the assumption. By this method | ^ dx, where -p. is a 

rational fraction, may always be reduced to the sum of a number of 
integrals of the forms 

( Adx f Bdx f {M + Nx)dx f (K+Lx)dx 

Jx-aJ ){x-by' Jx'-2ax + a' + ^'' ^^ ){x' -2ax + a' +f3J ' 

The two first of these are integrable by inspection, the third can 
always be integrated by the algebraical artifices before considered, 
and the last may be reduced to the third form by the method of 
reduction which will be investigated below. 

The general method of determining the partial fractions above 
given is often very laborious, and may be simplified in practice in 
the manner indicated by the following examples. 

72. Let F=0 have neither equal nor impossible roots. We may 
then proceed in every case as in the following example. 



INTEGRATION. 45 

J rUdx _ r xdx 

]-V--]i,a:-a){x-b)i,x-cy 

^, X ABC 

Then assume , ^7 -rr-, r = + 7 + , 

{x - a){x ~ o) {x - c) X — a x ~ o x — c 

.'. x = A{x- h) {x - c) + B (x - a) {x - c) + C (x - a) (x - h). (l ) 
Since this must hold for all values of x, 
let x = a, .'. a = A (a — b) (a — c), 

a: = b, .-. b = B(b-a)(b-c), 

X -Cj .'. c= C(c — a) {c — 6), 

which equations determine A, B, and C; and then 

C xdx _ fAdx CBdx f Odx 

J {x — a)(x-b){x-c) J x-a J x- b J x-c' 

= A log {x — a) + B log (x-b) + C log (x - c). 

73. If we had pursued the method of Art. 71, we should have 
determined the values of ^, B and C, by equating the coefficients 
of x^ and x and the constant terms of equation (1). This would 
have given us the equations 

= A+B + C, 

l=-A{b + c)-B(a + c)-C{a + h), 

= Abe + Bac + Cab, 
which give A, B, and C though less readily than the equations of 
the last article. ^ 

74. Let F= contain equal roots but no impossible ones, as in 

, f (x'^ + x)dx 
the example \r-^ — ^ t^. 

^ J (x- ay (x - b) 

x^ + x A2 A. B 

Assume z„y ^ = 7 ^ + + y j 

{x — a){x — b) {x — ay x — a x-b 

.-. x^ + X = A2{x^b) + A^{x - a) (x-b) ^ B {x - aY ... (a). 

Let x = a, .'. a^ + a=A<i{a-h) (1), 

therefore subtracting, 

(x''-a^) + {x-a) =A. (x-a) + A, (x-a) (x-b) + B (x - a)\ 
and dividing hy x — a, 

x + a + l=^A2 + Ai(x-b) + B(x-a) (/3). 

Let x = a, .: 2a + l=A2 + Ai(a-b) (2).* 

* This step is admissible although we have previously divided by a^ — a, because 
(u) and (/3) are not equations for the determination of a?, but identical equations. 
Thus when the proper values are given to A2, A^ and B both sides of equation (/3) 
become a: +a+ 1, and must therefore have the same value when x is put equal to a. 



46 INTEGRATION. 

B may be determined by putting x = b either in (a) or (/3) ; the 
former is preferable, as it gives B independently of A 2 by the 
equation 

b'+b^B{b-ay (3). 

Equations (1), (2), and (3) give ^2? ^i> ^^d Bj and 

'Bdx 



}{x-af{x-b) j{x-ay J x-a ja 



= -^ +A, log (^- a) +^ log {x^b) + a 

X — Cli 

The same method applies when there are more than two roots equal 
to «_, by repeating the substitutions and subtractions. 

75. Let V = contain unequal pairs of impossible roots, as in 

, . C (x -c)dx 

the examp\ej^^^~^^^^^~^^. 

x-c Ax + B Cx + D 

Assume 7-^ ^w-^ — -. rr. = —o ^ + 



{x''-\-a!'){x^-hbx + b') x^-\-a' x^+bx + b" 
.-. x-c = {Ax + B) {x^ -^bx + b') + {Cx + D) {x^ + a') . . . (a). 

Let x^ = -a^; an assumption which we are at liberty to make, 
since the above equation, being identical, must be true even when 
impossible values are given to x; 

.'. x-c = {Ax + B){bx + b'-a% 

= Abx' + {A(b'-a') + Bb}x + B(b'-a% 
= {A {b' - a') + Bb}x-Aba' + B (b'- a'), 
which cannot hold unless 

i=Aib'-a') + Bb (1), 

and -c = -Aba'+B{b'-a') (2). 

Equations (1) and (2) determine A and B ; C and D may be 
determined in exactly the same manner. 

Thus, in equation (a), let x^ = - (bx + b'), 

.'. x-c = {Cx + D) (a'-bx - b'), 

= -Cb x^+ {C {a' - b') -Db}x + D (a' - b'), 
= {C {b' + a' -b') - Db} X + Cb b'+ D {a'-b'l 
which cannot hold unless 

l = C{b'+a'-b')-Db (3), 

-c=Cbb'+D(a'-b') (4). 






INTEGRATION. 47 

Equations (3) and (4) determine C and J) ; and 

r {x-c)dx ( {Ax + B) dx f{Cx + D)dx 

]{x' + a')(x^ + bx + b')~J- a' + a"" "^J x^ + bx + b' ' 

These integrals can be found for all values of a, b, b' by alge- 
braical methods. 

When only one partial fraction remains to be determined, as in 
this example after A and B have been found, it may be done by 
substituting for the other indeterminate quantities in (a) their values, 
and dividing by the coefficient of (Cj: + D). This is generally a 
simpler method of determining the last partial fraction correspond- 
ing to a pair of impossible roots ; we shall employ in the following 
example, 

x^dx 



I 



{x-iy{x^+i)' 

x^ _ A^ A, Cx+D 

^^^"^"^^ (x-iy{x'+i)~{x-iy^x-i'^ x'+i ' 

.-. x' = A, (x' + 1) + A,(x-1) {x' + 1) + {Cx + D) {x -ly . , . (a). 

Let ar = ], .-. 1 = ^2.2, .*. A = i; 

therefore subtracting, 

x^-l=A2(x^-l) + A,{x-l){x^+l) + {Cx + D){x-iy, 
.'. x'' + x + l^A2{x+l)+A,(x' + l) + {Cx + D)(x-l). 

Let x = l, .-. 3 = ^2.2 + ^1.2, .-. A,= ^ =1; 

substituting these values in (a), it becomes 

'^-x + i = (Cx + D)(x-iy, 

.-. Cx-hD = i, or C=0, Z) = i, 

r x^dx _ 1 f ^^ f ^^ ^ f dx 

•'• j (x - ly {x' + l)~ ^J {a;~ ly '^ Jj^i'^ ^ Jsc^+1 

= -1. - + log(a:-l)+ Jtan-';?7 + C. 

X — I 

When there are more than two pairs of impossible roots, the- 
partial fractions may be determined in succession by the method 
given in the first example of this article, the last fraction being most 
conveniently found by substitution, as in the last example. 



48 INTEGRATION. 

76. Let V = contain equal pairs of impossible roots, as in 
the example j ^^^^^,^^,^^^ . 

^a^-x Ax + B A'x + B' Cx + D 

Assume ^^,^^^,^^,^^^-^p-^+ ^,^^ + ^, ^ ^ , 



= {Ax + B)(a^ + l) + (A'x + B'){a^ + 2){x^ + l) + (Cx+D){x' + 2y..X^)' 

Let a:' = -2, .-. --4>- x = {Ax + B) (-1) ... {f^), 

which cannot hold unless 

-1 = -^, or ^ = 1, 

-4 = -ig, B = 4>. 

Subtracting (yS) from (a) we have 
Qx'+4> = (Ax + B){x^ + 2)+(A'x + B'){x'+9)(x'+l) + {Cx+D)(x^-i-2y; 
dividing by x^ + 2, 

2 = {Ax + B) + (A'x + B') (x^+1) + (Cx + D) {x'-¥2) (7). 

Leta'= = -2, .-. 2 = (Ax + B)-{A'x + B') (3), 

.-. = A-A', .: A'=^l, 

2:=B-B', B' = 2. 

The remaining numerator, Cx + D, may be determined by put- 
ting 0?'= — 1, as in Art. 75, or, being the only remaining numerator, 
more conveniently by substituting for A, B, A\ B', their values in 
(a ). If there had been three or more equal pairs of impossible roots, 
we must have subtracted (B) from (7), divided by {x^ + 2), and then 
put 07* = - 2, and repeated the process till all were determined. 

The partial fractions being determined, we shall have 

r {2x'-x)dx _ f (Ax+B)dx f {A'x + B')dx r (Cx+D)dx 
J(x'+2y{x'+l)~J {x' + 2y "^j x' + 2 "^j af'+l ' 

The first of these integrals may be determined by the method of 
reduction, and the others by known methods. 

77' We have now considered every case except where roots of 
all the different kinds occur together, in which case the different 
methods must be applied in succession, as it is easily seen, by observ- 
ing the preceding examples, that the determination of any partial . 
fraction is not at all affected by the number or nature of the rest. 



INTEGRATION, 49 



Rationaliza lion . 



78. Many functions, which are not in the form of rational frac- 
tions, may be reduced to a rational form, or, as it is termed, ration- 
alized by different transformations. Rules for the rationalization of 
several classes of functions have been investigated, but the student 
will find that practice will enable him to discover the appropriate 
assumptions in most of the cases which ordinarily occur, without 
burdening his memory with a variety of rules for the purpose. 



Take as an exam 
Assume 



. f x^dx 
pH \-^ 1. 

J x-^ + a'^ 



,\ x^ = z^, 

xi ■= ^, 
and dx = 6z^dz, 

/z'^dz 
J , which is in a rational form, and 
Z^ + «2 

may be integrated by the methods already investigated. 

Integration of x'^{a + hx'^ydx. 
79. Functions of the above form are of very frequent occur- 
rence. We shall first shew how to integrate them when the indices 
7w, w, and p satisfy certain conditions, and then investigate a general 
method of integration applicable to all cases. 

(1) Where is a positive integer, m, n, and p being posi- 
tive or negative, integral or fractional. 

7* 
Let p — - , where r and q are positive or negative integers. 

Assume (a + bx"") = z'^, 

z'i-a 



.-. X" 



b ' 



'. X 



m+l > 

b~^ 



nb " 



H. D. C. 



50 INTEGRATION. 

and multiplying by c*^, we have 

C f m+\ _ 

j x-{a + bxjdx = -^ j z-'-'-'iz^ -«}""' dz. 
nh " 

Now when the above condition is satisfied 1 must either 

n 

equal zero or a positive integer, and the function under the integral 

sign can be expressed in a finite series of integral powers of z, and 

can therefore be integrated in finite terras. 

If had been negative or fractional, the expanded binomial 

would have contained an infinite number of terms, and the method 
would have failed to give us the integral in a finite form. 

The condition = a positive integer, is called the First 

Criterion. 

(2) Where + p = a negative integer. 

Then x^ {a + hxy = ^"^"^ {b + aa?-"p. 

The function in this latter form may be integrated by the 

previous method if ^ = a positive integer ; that is, if 

+ p = a negative integer. 

The assumption to be made in this case is therefore 

r 
b + aar~" = z', putting p = - as before. 

The condition + p = a. negative integer, is called the 

Second Criterion. 

80. The following are examples of the application of these 
methods. 



r x^dx 



(«2 + a?2)s 

Here = "^ — = 3, a positive integer ; therefore the first 

criterion is satisfied. 

Since p = — ^, the proper assumption is 

a'2 + x'-i = z'. 



integration; 1 

Here = — , and the first criterion is not satisfied. 

n 2 

But + p = -|+i = -2, and the second criterion holds. 



Then 



J x' ~J x' 



and we must assume a^x ^ — 1 =2,% 

[{a'x-'-iy^dx 1 r 2/ 2 -.N^ 

15aV '■^• 

81. When neither of the criteria is satisfied, we must employ 
the following method, which is applicable to all cases. It consists 
in making Jo:'" (a + 6.r")^ (/a; depend on another integral of the same 
form but with smaller values of ?w or p ; by repeating which process, 

4 — 2 



52 INTEGRATION. 

we arrive at last at an integral which can be determined by methods 
already given. This is called the method of reduction. 

As the treatment of the integral depends on the signs of m and /?, 
we shall use these letters to represent positive quantities only, which 
will give the following cases. 

r x*^dx C 

(1) To reduce m in I ^ ^^„. ^ or Xx'^ia^hx'J dx. 

, . rr, , , ( dx f(a + bx"y , 

(2) To reduce m m j^^.7(^^^- or j'^-^^ dx. 

(3) To reduce p in \ x"^ (a + hx^ dx or /^ ^^^ dx. 

(4) To reduce p in /^^|^ or f ^„^/^\^y 

The mode of treatment is exactly the same in each pair of integrals 
as above arranged, that is, the method of reducing m or p depends 
only on the sign of the index to be reduced. 

n is always supposed to be positive, as the function can always 
be reduced to that form by altering the value of m. The follow- 
ing are the methods to be employed in the four cases respectively. 

• P - f:c'"-"+' X"-^dx _ r n»-=+i 7 / ^ ) . 



therefore, integrating by parts, 

m — ?i + 

+ 



■ 1 f x--"dx 

l)}{a + bx-r'"'^^'>' 



nb{p-\)(a + bx"/-' nb (p 

x"^"""' m-n + 1 f x"^"{a + bx'']dv 

"•" ?ib (p - 1) (a + bx""/-' ^ ?ib {p-i)J {a + bxy ' 

p^ x^-"^' m-n + l ^ 

nb {p-\) {a + bxy-' "^ w6 (p- 1) ^'^ i^„._« +6i"4 ; 



" " b{np-m-l)^a + bjr)^''^b{np-m-i)'' "'-"' 

by which formula P^ is made to depend on P,„_„, that is, 

f x'^dx r x'"~"dx 

Jia + bx-y''^ J{a-^bx'y" 

and by repeating the process, P„,_„ may be made to depend on 



INTEGRATION. 53 

P^_2„, and generally P,„ may be made to depend on P,„_r„ where 
r is any integer. 

By employing the form (A) we make the integral depend on 
another of the same form, in which m is diminished by w, and p by 
unity, by the repetition of which m may be reduced by any multiple 
of n and p by the same multiple of unity. 

I x'^{a-\-hx'y is treated in exactly the same manner, and we obtain 

formulae by which m alone may be reduced by any multiple of w, 
or m reduced by a multiple of n, and p increased by the same 
.multiple of unity. 



then 



83. LetP, 

1 



C dx 

Jx"'{a + bxy ' 



x"" {a + hx")^^ x"" (a + bx'^y x'^'* (a + bxy * 

'*• "^^^ " j x^'ia + bxy-' " ^^"-" ' 
and 

j x^{a^bjry-' " j(a + bx'^y-^ ^i {m-\)x'^-'\ ' 

1 nb{p-\) r dx 

On-1) x""-"^ {a + bxy-' m-l j j?*^" (a + bxy 
by integrating by parts : 

. „p 1 n(p-l) + m-l ,j, ^ 

by which formula P,„ is made to depend on Pw_„, and therefore 
P„ may be made to depend on an integral of the same form, in 
which m is reduced by any multiple of n. The other integral in (2) 
is treated in the same manner. 

84. Let Pp = r ^"^ (a + bx'^ydx ; 

.-. P^^a j x'"{a + bxy-' dx + b f ^•"+" (a + bx")^-' dx ; 

and [ x"^" (a + bxj-' dx = j x^^' d |^^±M'| , 

x'''^' {a + bxy m + 1 [ ^. , , . 

by integrating by parts : 

^ ^ np np ^^ 



54 INTEGRATION. 

^ ??p + m + 1 7<J9 + 7W + 1 ^ 

by which formula Pp is made to depend on Pp_i, and therefore 
Pp may be made to depend on another integral of the same form in 
which 2^ is reduced by any integer. The other integral in (3) is 
treated in the same manner. 



85. hei P,= [j-^^^^^^; 
^ J (a + hxy 



then 



{a + hxy 
x"^ ax"^ bx" 



{a + hxy-'' {a + hxy {a + bx^f ' 

""""^ j (« + bxy~ j "^ "^ l«6 (p - 1) (a + bx")^'j 

^~ nb(p-l){a+bx"Y-' '^nb(p-l) J {a + bx^-' ' 
by integrating by parts : 

p j^;;;^; f m+i ) 

•• ""^"w (p -!)(« + 6.T"y-' U(p-l) J "-" 
by which formula Pp is made to depend on Pp_i, and therefore 
Pp may be made to depend on an integral of the same form in 
which p is reduced by any integer. The other integral in (4) is 
treated in the same manner. 

86. Of these four forms the first two reduce m alone by n at 
each step, and the last two reduce p alone by 1 at each step, while 
the form (A) which occurs in Art. 82 reduces both m and p at once, 
and a similar form reduces 7n and increases p in the other integral 
of Art. 81, (1). On account of the increase of p, this latter form is 
never more advantageous, and often much less so, than the final 
form of Art. 82. Where however it is equally applicable, it has the 
advantage of being more readily obtained. The mode of proceeding 
with an integral of the general form under consideration, will be to 
reduce m and p together by (A), where that is possible, until either 
m is less than ?^, or p equal to unity. If the function is not then 
integrable, we must reduce p in the former and m in the latter case, 
until we arrive at an integrable form. Where formula (A) is not 
applicable we must begin by reducing m or p, choosing that one by 



INTEGBATION. 55 

the reduction of which we arrive most easily at an integrable form. 
That we shall always obtain at length an integrable form is evident, 
since we may always reduce p to unity, either in the numerator or 
denominator, when the function is integrable at once in the first 
case, and in the second is either a rational fraction or admits of im- 
mediate rationalization. 

The reduction of m is often preferable to that of p, because it 
proceeds by n instead of unity at each step, and in many cases, 
as where m = rn — l, leads to a function which is integrable by 
inspection. 

The methods employed in the four distinct cases, aiid not the 
resulting formulce, should be remembered and applied in any par- 
ticular case. 

87. We will take as an example / -^ . 

Here we can first reduce 7 by 2 and f by 1, by means of for- 
mula (A), and then reduce the index of x alone by 2, twice, which 

will bring us to I — j which can be immediately integrated. 

Employing therefore the method of Art. 82, we have 

x^ ^ f x'dx 

= r + 6 i . 

{a' + xy Jin' + xy 

As we have now to reduce the index of x twice, we will put it 
equal to m, that the same process may serve for both reductions. 

Let then p,,.= f_f!i^; 

.'. (as in Art. 82) P„ = x"^-' (a' + x')i-{m-l) L"'"^ (a' + x')'^ dx 
= X-' {a' + x'f -{m~l) {a' P„._, + P,4, 

and P^ = lx\a' + x')l-la'P,, 



56 INTEGRATION. 



TT I x''clx 

Hence 



, „ f ocdx . ^ „.i 

d P, = J = {a? + x^y + const. 

r x'dx _ x^ 

Jia' + x')^'" {a' + x'f^ 

+ 6 [^x' («' + ^^)2 - 4-a=^ {i-x' {a' + x"-)^-ia' (a' + x^)i} + const.] 

88. There are some particular values of the indices p, m, and ?ij 
in which the above methods become inapplicable on account of the 
coefficients which occur becoming equal to infinity. This happens 
in the following cases : 

(1) Where the index of a; is (—1) the reduction of m is im- 
possible, as appears from Art. 83. The integral is then of the 

/* dx C dx 

form I — j—^ or — (« + hx'^y. The most general form of p 

J X \Ci T" ox J /J? 

' '*' 1 , . ... 

IS -, r and q bemg positive integers. 

Assume then a + hx'^ = z''i 

(z^ - ay 

.*. X = T } 

b" 

dx q z'^~^dz 
X n (z'^ — d)' 

and the integrals become 

n J z^ — a n J z^ — a 

both of which are integrable as rational fractions. 

,^ „,, m + 1 . f x"'dx f(a + bx"ydx 

the reductions of m and p respectively become impossible, as appears 
from Articles 82 and 84. 



r x'"dx _ f x"^"Pdx 
^"^ j {a + bxy~ J (a^-"+AV 



=/ 



by 

dx 



x{ax-''+by 



by the above condition. 



INTEGRATION, 57 

This is the same form as that just integrated, and may therefore 

V 

be solved by the assumption aa;~" + J = s' if p = - . 

(3) When the index of {a + ia;") is (- 1) the reduction of p 
becomes impossible, as appears from Art. 85. In this case the in- 

r^^ d.x f dx 

fractions. 

Integration of siif'x cos"xdx. 

89. This function may be integrated by methods similar to 
those applied to x"'(a + bx''y. We will first consider some particular 
cases in which the integral may be immediately obtained, and then 
proceed to the method of reduction. 

(1) Let one of the indices (as m) be an odd positive integer, 
then m = 2r + 1, where r is a positive integer ; 

.*. / sin"*a? cos"j:c?:c = / cos"j: (1 — cos^o?)'' sin xdx. 

By expanding the binomial, the integral is reduced to a finite num- 
ber of integrals, each of which may be determined by inspection. 

If the index of cos x is an odd positive integer, and equal to 
2r + 1, we have 

I sin'"j7 cos'^xdx = I sin'"a; (1 — sin^j:)'' cos xdx, 

which is integrable as before. 

(2) Let (m + n) be an even negative integer =— 2r. 

Then / sin'^o: cos"a? dx = I tan'^o: cos"'+"^ dx 

= I tan*"^ (I + tan^cr)''"^ sec^j: dx. 

By expanding the binomial, the integral is reduced to the sum of 
a finite number of integrals, each of which may be determined by 
inspection. 

90. When neither of these conditions is satisfied we must pro- 
ceed by reduction. 

For this there are only two distinct modes of proceeding, that 
for the reduction of a positive index, and that for the reduction of 
a negative one. 



58 INTEGRATION. 

(1) To reduce m in lsm'"x cos" jt^j?, 

being positive, and n either positive or negative, let 
P,„ s= / sin"* X cos" X dx; 

Then P,„ = I sin"""^ x cos" a; sin x dx, 

sin"'-^jrcos"+^a: m-\ [ , ^. „^, , ,.. 

= + / sin"'-2j: cos"+'a;c?ar (A), 

w + 1 n + 1 J ^ / 

sin'"-'j;cos"+^ar m-l ,[ . ^. „ , „, 

= + { / sm"*-^ j: cos"a: dx - P^}, 

w + 1 w + 1 7 ^' 

_ sin'"-*a;COs"+'a: vi-1 

If the index to be reduced had been that of cos ar, the method would 
have been exactly analogous. By formula (A) both indices are 
altered ; that of sin x reduced, and that of cos x increased or reduced 
according as it is positive or negative. 

X N m 1 . (sAif'xdx 

(2) To reduce ;i in j ^^^„^ , 

n being positive and m either positive or negative, 

, „ {s\n'^xdx 

let P„ = ;i— ; 

; cos" J? 

_ /"sin'"j:(cos^j: + sin^j:)rfa: 
" ~ j cos"^ ' 

= Pn-2 + / sm'^+'a; — , 

I cos" a: 

_ sin'"+^a; m+l fsirfxdx 

"-"- ^ (7i-l)cos"-'^ ~ 11 -\ J cos"-'a.- ' 

sin"'+^3: j m + ] \ 

~ (n-l) cos"~* a; ( w - 1 / ""■^* 

If the index to be reduced had been that of sin x, the method 
would have been exactly analogous. 

It will be seen that in every case the reduction is by 2 at 
each step. 



91. Take as an exam 



, f dx 
P^e — 5-- 

J COS^ J 



I 



INTEGRATION. 59 

Let P„ 



f dx 
jcos"^;' 

-/ 



(cos'ar + ^m^x)dx 
cos" a: 

sin ar^j; 



I- 



P^2 + I sin a: 



_ sin J? 1 f dx 

~ "-' "*" (»i-l)cos"-'a? ~ 72^ jcos"-'cr' 



_ sin a: w - 2 p 

•• ^~4cos*a:"^4 "' 
sinjr 1 



2 cos^ ir 2 



r dx 

/■ r/o? sin or 3 sin a: 3. r^ .^f\ i \ 

jcos'o; 4cos'ar 8 cos* a: 8 ° ^"^ ^ ^ 



CHAPTER V, 

SUCCESSIVE DIFFERENTIATION. 

Theory of the Independent Variable. 

92. In Chapter III. we have shewn how from any equation of 
the form 

y=f{x\ or ^ = </)(j/) (1), 

to determine the value o? f'{x) or (p'{y) in the corresponding dif- 
ferential equation 

dy =f\oD) dx, or dec = (p' (y) dy (2). 

It may now be asked, are the differentials dx, dy to be con- 
sidered as functions of tlie variables a; and y ? The answer obviously 
is, that they may have any values consistent with equations (2), that 
is, any values whose ratio equals the differential coefficient. Now 
these values may evidently be such as to make them both functions 
of the variables; but we are at liberty also, if we please, to give to 
either of them any arbitrary constant value, when the other will 
receive such a corresponding value as to satisfy equations (2), and 
will therefore be a function of the variables. 

We have already said that a single equation between two variables 
is called an equation of one independent variable, without applying 
the term independent variable to one rather than the other ; we will 
now add the following definition. 

Dep. In an equation between two variables, that one whose 
differential is assumed to be constant is called the independent 
variable. 

When no such assumption is made and the differentials are left 
in their original generality, the equation is said to have a general 
independent variable. 

When y is given explicitly in terms of x, it is generally most 
convenient to make x independent variable, and vice versa, as 
thereby equations (2) are presented in a more advantageous form. 
It is sometimes, however, better to keep the independent variable 
general for the sake of symmetry in equations derived from equa* 
tions (2) in a manner which will presently be explained. 



SUCCESSIVE DIFFERENTIATION. 



61 



y 
I 






^T / 


/ 

s 




X 




'^l 




V 




/ 







M N L 



93. The following geometrical illustration will make the last 
article more intelligible. 

Let P, Q be any two points of 
the curve corresponding to equa- 
tions (1); TySy any points arbitra- 
rily chosen in the tangents at P 
and Q. Then, the remaining lines 
of the figure being drawn parallel 
to the axes respectively, TV and 
TV may be taken to represent the 
values of the differentials at P, and 
QZ and SX to represent their values at Q. 

The positions of T and S being perfectly arbitrary and inde- 
pendent of one another, the magnitudes of the differentials at Q are 
quite independent of their magnitudes at P. But we may if we 
will so fix the positions of T and S that W and QX shall be con- 
nected by any law we please. 

Thus we may impose the restriction that one of the differentials, 
as dx, shall at every point of the curve be some determinate function 
of the corresponding abscissa, or ordinate, when dy will also take a 
variable value dependent on the assumed value of dx and on the 
form of the curve. Suppose for instance we determine that dx 



shall always have the value — , i. e. that T, S, &c. 
chosen that 



shall be so 



PV= 



OISP 



and QX== 



ON' 



MP """ ^^^ ' NQ ' 
and that the same law may prevail throughout the curve. 

The corresponding values of dy will then be given by the 
equations 



TV= 



OM' 



.tanTPr, SX 



ON' 



tan SQX, 



MP ^^'' ^" NQ 

By this assumption both dx and dy are made functions of the 
variables, their analytical values being 

djo = j, dy = -f{x). 

In like manner we may determine the positions of the points jT, 
'S', &c., throughout the curve, by any other rule, or we may leave 
them if we please, perfectly arbitrary and unfettered by any rule at 



62 SUCCESSIVE DIFFERENTIATION. 

all, since the equations (2) are equally true whatever rule is adopted 
and whether any rule be imposed or not. 

One of the simplest rules for fixing the values of the differentials 
is to give PV the same value at all points of the curve, so that 
QX= PV wherever P and Q may be. In this case TF will not 
preserve a constant value, since TV= PV .tan TPF and the angle 
TPF varies from one point of the curve to another. In fact we shall 
have with the above rule 

dx=C, dy = Cf{x). 

Another equally simple rule is to make TV constant throughout 
the curve, that is, to assume 

dy = C and thence dx = C^'{y). 
When PV retains a constant value x is (as above stated) called the 
independent variable, and when TV is constant y is called the inde- 
pendent variable. 

Another form of restriction sometimes imposed upon the differen- 
tials is to take some function of one of the variables as -v^ (x) and 
make its differential ^'{x) dx constant, that is, to determine dx and 
dy by the equations 

When this is done, ^{x) is said to be the independent variable. 

When no restriction is imposed on the values of PFand TV, the 
independent variable is said to be general. In this case therefore 
the differentials must be treated as arbitrary functions of the vari- 
ables, to which we can at any time assign such definite forms as may 
be found convenient. 

94. Since dx^ dy are in general (before any variable is made the 
independent variable) functions of the variables, they, like all other 
functions, may have differentials, and we shall meet with such quan- 
tities as d{dx), d{dy), d{d{dx)}, d{d{dy)}^ &c., which are written for 

consciseness 

d^Xj d^y, d% d^y . . . d^x, d'y, 

and are called the second, third . . . w'** differentials of x and y re- 
spectively. 

Again, yX"^) being a function of x must have a differential co- 
efficient, and this a third, and so on. These are written 



SUCCESSIVE DIFFERENTIATION". 63 

and are called the second, third . . . n^^ differential coefficients of 
/(.r) or 1/ with respect to x. So also we have 

f"(j/)' 0"'W •••*"«> 

the second, third ... w*^ differential coefficients of (/)(y) or x with 
respect to^. 

95. It should be observed that the idea of a particular inde- 
pendent variable applies only to differentials and not to differential 

coefficients. The ratio -p or /'(a?) is the same whether dx or di/ be 

variable or constant and is in no way affected by the assumption of 
a particular independent variable. So from the figure of Art. Q3, 
it is evident that tan TPF is independent of the position of T. 

Again, f'\x) which = ^y' has the same value whatever be the inde- 
cix 

pendent variable, and the same is evidently true of all the successive 
differential coefficients. An equation among differential coefficients 
therefore, if it involves no differentials, does not imply the assumption 
of any particular independent variable and is equally true irrespective 
of any such assumption. It will be seen hereafter that an equation 
among differential coefficients with respect to x, is immediately con- 
vertible into a corresponding equation among differentials with x for 
independent variable^ In consequence of this, x is sometimes incor- 
rectly called the independent variable in the former as well as in the 
latter equation. It is important that the student should avoid this 
error*, as some of the following propositions wholly rest upon the 
fact that an equation among differential coefficients is unaltered by 
a change of the independent variable. 

96. When f\x) or 0'(y) is known, the ratio q^ dy to dx is also 
known. So when /'(a;) and/''(.r) or (l)'{x) and 0"(.r) are known, we 
can find the relation between dx^ d^x, dy and d^y ', and generally 
when the first n differential coefficients of one variable with respect 
to the other are known, we can find the corresponding relation among 
the differentials of the variables of the orders up to the w**". These 
relations we shall now determine. 

* The phraseology referred to is called erroneous, with reference only to the defini- 
tions before given. Totally different definitions are sometimes employed which 
would justify that application of the term ' independent variable.' 



64 SUCCESSIVE DIFFERENTIATION. 

97. To find the relations between the successive differentials 
and differential coefficients when the independent variable is general. 

Let y=f{x), 

then dy=^f{x)dx (1); 

differentiating both sides of this equation^ we obtain 
d'y = d{f{x)dx} 

= df\x) dx tf{x) d'x, Art. 23. 

.-. d^y=fXx)dx'+f(x)d^x (2). 

By differentiating this equation, we obtain (Art. 21 and 23), 
d'y = dfXx) dx' + d (dx')fXx) + df'{x) d'x + d^xfix), 

... ^y= f"{x) dx^ + ^f"{x) dx d'x +f'{x) d^x (3), 

and similar equations may be found connecting the differentials and 
differential coefficients of higher orders. 

These equations give the successive differentials of y explicitly 
in terms of the differentials of x and the differential coefficients of y 
with respect to x. The differential coefficients may be found expli- 
citly in terms of the differentials from the above equations, but more 
readily as follows. 

98. To express the differential coefficients in terms of the dif- 
ferentials when the independent variable is general. 

If 3^=/W, 

/'w=l (*)' 

...(A...6)/"« = j,.{|} = ^-5^^'^^^^ (5). 

A.a.,r(.)=^=i,.{±.(|)} 

_ dx{dxd^y - dydPx) - S d'x{dxd^y - dyd^x) 

~~~ " dx' ^^^' 

and similar equations may be found for the differential coefficients of 
higher orders, the equations (4), (5), (6), &c. being equivalent to 
(1), (2), (3), &c. and deducible from them. 

99. To find the relations between the successive differentials 
and differential coefficients when x is independent variable. 



SUCCESSIVE DIFFERENTIATION', 65 

When -r is made independent variable dx is constant, and therefore 
d^x, d^x, &c., all equal to zero. Hence we obtain by differentiating 
the equation, 

.-, d^y=^f"{x)dx^y since dx is constant, 
d^y=f"[x)dx\ 



dry=f^''){x)dxr\ 






equations which might have been obtained from those of Arts. 
97 and 98 by equating d'Xj d^x, &c. to zero. 

100. Examples of successive differentiation. 

1. To find the successive differential coefficients of y where 

We have by repeated applications of the rule of Art. 27, 
f{x) = n . ax"-\ 
fX^) = n.n-l.ax"-% 
f\x) = n.n-1...2.1.a, 

* From the above equations we see that when a; is independent variable the 

successive differential coefficients may be written as above -~ —...— ^. These 

dx dx^ dx'* 

expressions are sometimes used instead of /' (a?), f"{x)j &c., when x is not inde- 
pendent variable, but in that case it is clear that the numerators will no longer 
represent the successive difterentials of y, and that these expressions in fact cease to 
be fractions, and become mere symbols equivalent to f(x)j f"{x)...J^"^(x), Thus, 
if this practice were adopted, equation (2) of Art. 97 would become 

the d^y on the left of the equation, meaning the second differential of y, while in the 

d^y 
numerator of -r-^ it has no such meaning. It is better to avoid such a use of the 

notation, but as it is often met with, it must be remembered that -tK , &c. are then 

dx-' 

mere symbols and not fractions. 

H. D. C. 5 



66 SUCCESSIVE DIFFERENTIATION-, 

and all the differential coefficients after the 7^*** vanish, since f"{x) 
is constant. 

The successive differentials when x is independent variable follow 
at once from the differential coefficients : as appears generally in 
Art. 99. 
Thus we have 

d^ =f'{pc) dx -n. ax'^'^dx, 
^y =f"{x) dx^ = n.n-l. ax^'-'dx'. 



dy =/"'(ar)cZa;" = w . w - 1 . . .2 . 1 .adx\ 
If it is required to find the values of dt/j d^y^ &c. when the inde- 
pendent variable is general, we must proceed exactly as in the 
general case of Art. 97* Thus, by differentiating repeatedly and 
considering dx variable, we obtain 
y = ax^, 

dy = n, ax""'^ dx, 

d''y = n.n-l,ax"-''dx^ + n.ax"-'d% 
d^y = n.n-l,n-2. ax^'^dx^ + n.n-l. ax"-^ {2dx d^x) 

+ n.7i—l. ax^~^ dxd^x + n . ao:""^ d^x, 
= w.w-l.w-2. ax'^'^dx^ -hSn.n-l. ax^-'^dx d^x + n. ax"-^d^x, 
and by pursuing the same method the higher differentials may be 
found. 

2. Let y =f{x) = ^. ' 

Then proceeding as before, /X-^) = ^% 

and when x is independent variable, 

dy - e^'dxj 
d^y^edx^y 

d"y = e'dlr". 
When the independent variable is general, we have 

y = ^% 

dy = e^dx, 
d'y = e^dx- + e'd^x^ 



SUCCESSIVE DIFFERENTIATION. 67 

d^y = e'dx^ + ^,2dx(Px + edxd?x + ed'^x, 

= edx^ + S^dxd-x + ed^Xy 

&c. = &c. 

Further examples of successive diiFerentiatlon will be found in 
Gregory's Examples, Chap, ii. Sect. 1. The quantities there obtained 
are successive differential coefficients expressed by the notation men- 
tioned in the note to Art. 99 ; or if x is considered to be the inde- 
pendent variable dy, d^i/ . . . &c. properly represent the successive 
differentials. The student is recommended to work a few of Gre- 
gory's examples also with a general independent variable. 

. 101. We have already employed the term differential equation 
in speaking of equations involving differentials. For convenience 
sake, equations involving differentials or differential coefficients up to 
the 1st, 2nd. . .n^^ respectively, are called differential equations of the 
1st, 2nd.,.n^^ order. Thus the equations (1), (2), (3), of Art. 97 are 
differential equations of the 1st, 2nd, and 3rd orders respectively. 
Equations which (like those last mentioned) are obtained imme- 
diately by successive differentiation, are also sometimes called the 
1st, 2nd, 3rd, &c. derived equations, and the original equation (as 
y=f(x) in Art. 97) is then styled the primitive. Thus in the 2nd 
example of the previous article the last equation is the third derived 
equation of the pririiitive y — e*, and is a differential equation of the 
Srd order. 

Besides the derived equations which are the immediate results of 
differentiation, an infinite variety of differential equations of every 
order may be formed by combining together the primitive and its 
derived equations in any manner we please. 

Every differential equation, again, whether an immediate derived 
equation or compounded out of several, may be presented in a 
number of different forms by adopting a general or different particu- 
lar independent variables. Thus the derived equations of Art. 97 
assume the simple forms of Art. 99, when x is made independent 
variable. By altering the independent variable we vary only the 
form and not the substance of a differential equation. The form 
corresponding to a general independent variable includes all the 
others, and we shall shortly prove that any one of the particular 
forms may be deduced from any other. Before investigating the 

5—2 



68 SUCCESSIVE DIFFERENTIATION. 

methods of effecting such transformations, we will give some exam- 
ples of the formation of differential equations by combinations of 
various primitives with their derived equations. 

102. Let the primitive be 

y = ax-\- hx^ (1). 

The first two derived equations (when the independent variable is 
general) are 

dy = adx+ 2bxdx (2), 

d^i/ = 2bdx'' + {a + Ux)d'x (3). 

1 . Let it be required from these equations to obtain a differential 
equation of the first order, in which a; shall not appear. 

To do this we have only to eliminate cc between (1 and (2). 
Thus, from (2) we have 

df = (a' +4>abx + 4i V) dx^ 
^ (a' + 4.bi/)dx' by (I), 
which is the required equation. 

2. From the same primitive let it be required to find a differen- 
tial equation from which a and b shall disappear. 

For this purpose we must employ equations (1), (2), and (3), as 
there are two quantities, namely a and 6, to be eliminated. Thus, 
eliminating a between (1) and (2), we find 

xdy — ydx = bx^dx ; 
and between (1) and (3), 

xd'y - yd^x = 2bxdx^ + bx^'d^x, 
and eliminating b between the two last equations we obtain the 
required differential equation 

x^d'ydx - ^x^dydx"" - x^dyd^x + '^xydx' = (4). 

If we had worked with x for independent variable we should have 
had, instead of equations (1), (2), (3), 
y — ax-\- bx^, 
dy = adx + 2bxdx, 
d'y = 2bdx\ 
By eliminating a and b between these, we find 

x^d^y - 2xdydx + Qydx"" = (5), 

"which differs from equation (4) only by the omission of the term 
involving d^Xj which vanishes when x is independent variable. 



^SUCCESSIVE DIFFERENTIATION. 6? 

. If we divide the last equation by rfo;* it becomes 

which by Art. 99 is equivalent to 

x'f{x)-<Zxf{x)-v<^y = (6), 

(4), (5), (6), are three different forms of the required equation. 

103. Equations formed as in the last example by eliminating 
constants are of frequent occurrence. In general, the order of the 
resulting equation must be equal to the number of constants elimi- 
nated. For to eliminate n constants we require in general w + 1 
equations. The primitive and its first n derived equations must 
therefore be used, and the equation thence obtained will involve an 
w*^ differential or differential coefficient, that is, it will be of the «"' 
order. In particular cases, however, it may happen that the steps 
necessary to eliminate n constants cause others also to disappear 
when the order of the differential equation will be lower than that 
given by the general rule. 

104. In some cases also we can eliminate functions of the 
variables as well as constants. 

Thus, let the primitive be 

^ y-a%\w{x-\-h) (1). 

i^'rom this we may eliminate the circular functions, as well as the 
constants a and h. 

The first two derived equations are (making x independent 
variable) 

dy = a cos {x + h)dx (2), 

d^y=-a sin {x + b) dx^ (3). 

Between (l) and (3) we may eliminate a sin {x + b) and obtain 
d''y^ydx^=0. 
If we had expressed this in the notation of differential coefficients 
it would have been 

f\y)+y = 0, 
and if we had kept the independent variable general, 

d^y dx — dyd^x + ydx^ = 0, 
as may easily be verified. 



70 SUCCESSIVE DIFFERENTIATION. 

Examples of the formation of differential equations by the elimi- 
nation of constants and functions will be found in Gregory's Ex- 
amples, Chap. IV. The results with a general independent variable are 
not there given, but if obtained, their accuracy may be tested by 
making d^x, d?x, &c. vanish when they ought to reduce themselves 
to the given forms. 

105. If we consider the w^^ power of a first differential as 
homogeneous with an n^^ differential, it will be seen that all the 
differential equations we have yet met with are homogeneous in the 
differentials. Thus, in this sense, every term of equation (4) in Art. 
102, is of three dimensions, and every term of equations (5) and (6) 
of two dimensions and of no dimensions, respectively. 

This homogeneity is not accidental, but must be found in every 
differential equation. For all such equations are either derived 
equations or are formed out of them. Now all derived equations are 
included in the general forms of Art. 97, which are homogeneous in 
the above sense: the same property must therefore belong to all 
particular derived equations and to all equations deduced from them. 

106. It has been before stated (Art. 101), that the different 
■forms assumed by a differential equation when expressed in terms 

of differentials with various independent variables or of differential 
coefficients may be deduced one from another, so that when any one 
form is given all the others may be obtained. This we shall now 
effect in the several cases. 

107. Having given an equation among the differential coefficients 
of y with respect to x, to obtain the equivalent equation among the 
differentials of x and y when x is independent variable. 

To do this we have only to substitute for f\x), f"{^) - • -/^"■'i^) 

in the given equation their values -^" , ~ . . . ~ from Art. gg, and 

if we wish to get rid of fractions, multiply by the highest power of 
dx which appears in the denominators. The converse transition is 
equally easy. For an equation among the differentials when x is 
independent variable can only contain the differentials dx, dt/,d'y...d"i/. 
By the equations dy =f{x)dx, d'y =f'{x) dx^-.d^y =f"\x) dx" of Art. 
99, we can eliminate the differentials of y ; when dx will be the only 
differential remaining, and since the equation must be homogeneous 



SUCCESSIVE DIFFERENTIATION. 7| 

(Art. 105) dx must enter to the same power in every tetm and may 
be divided out, leaving differential coefficients only. 

108. Having given an equation among the differentials of x 
and 1/, with a general independent variable, to find the equivalent 
equation where x is independent variable. 

To do this we have only to make dx constant, and therefore 

d^Xj d^x d"x equal to zero, when the equation is immediately 

reduced to the required form. 

109. Having given an equation among the differentials of x and 
^, in which x is independent variable, to find the equivalent equa* 
tion when the independent variable is general. 

The given equation may be transformed into that among the 
differential coefficients of y with respect to x, by dividing by some 
power of dx. (Art. 10?.) 

This may then be transformed into the equation among the 
differentials when the independent variable is general, by the fol- 
lowing equations, obtained (as in Art. 98) by the successive dif- 
ferentiation of the equation 
.f(x)=y, viz. 

„. X _ 1 , f 1 7 (dySX _ dxidxd^y - dyd^x^ - Sd'x (dxd^y-di/d^x) 
J ^''^'"dxXlx'^Xdx)]- '' Jx' ' 

&c. ■=^ &c. 
dx having been considered variable in the diflferentiations because 
the independent variable is general. 

The expanded forms after that for f'{x) are difficult to remem- 
ber, and should be worked out from the first or unexpanded forms 
in each particular case by performing the operations indicated, re- 
membering that both dx and dy are to be considered as variables. 

no. Having given an equation among the differentials of x 
atid y when the independent variable is general, to find the equi- 
valent equation when any function of j: or 7/ is independent variable. 

In this case, since neither dx nor dy is to be made constant^ the 
equation is still true in its original shape ; but it may be simplified 



72 SUCCESSIVE DIFFERENTIATION-. 

by eliminating one of the variables, and so obtaining a new equation 
between the differentials of the other and of the new independent 
variable. This may be done as follows. 

Let z be the new independent variable, and let x = (p (z), where 
(p is known. Then differentiating this equation, remembering that 
z is independent variable, we obtain 

dx — ^'{£)dzj 
d^x = <t>"(z)dz% 



d"x=(p^"\z)dz''; 
by substituting which values in the original equation we obtain the 
required equation between the differentials of ^ and z. 

By dividing by some power of dz, this may be at once trans- 
formed into the equation among the differential coefficients of ^ with 
respect to z, (Art. 107). 

111. In the preceding cases, one of the two forms of the equa- 
tion, either that to which or that from which we have had to pass, 
has had a general independent variable. In those that follow, both 
equations have determinate independent variables. Such transfor- 
mations are of much more frequent occurrence than the former ones. 

112. Having given an equation among the differentials of x and 
^ when X is independent variable, to find the equivalent equation 
among those of i/ and z when z is independent variable, z being a 
function of one of them as x, or as it is commonly expressed, io change 
ike independent variable from x to z. 

By dividing by some power of dx, the original equation may be 
transformed into the equation among the differential coefficients of ^ 
with respect to x. (Art. 107). 

Then by differentiating /(j:)=^, remembering that, when z is 
independent variable, dx is variable, we obtain 

/'W=l' 

/•r, M -Id (^y\ - d'ydx-d'xdy 

J ^""^-dx^dx) ~ 5? ' 

^//// ^ ^ ^J ^ ri(^y\\ dx{dxd^y - dyd^x) - U^x{dxdPy -dyd^x) 

J ^''^-Tx'^Xdx'^WxlS ' dc^' ' 

&c. = &c. 
Also, if the given relation between x and z is put in the form 

* = ^ (^), 



SUCCESSIVE DIFFERENTIATION-. 73 



we have dx = <l>' (2) ds. 



d''x = (p^"\z)d2\ 
when 2 is independent variable. 

By substituting these values of x and its differentials in the 
former equations, f\x), f'\x\ &c. may be expressed in terras of z, 
dz and the differentials of ?/, and the required transformation effected 
by the substitution of these values in the original equation. 

By dividing by some power of dz, this equation is transformed 
into the equation among the differential coefficients of ^ with respect 
to z. (Art. 107). 

113. Instead of first expressing the differential coefficients at 
length in terras of the successive differentials of x and ^, and sub- 
stituting for those of x their values derived from the equation a? = (2:), 
it will be found more convenient in practice to invert the order of 
these operations, and first substitute for dx its value in the unex- 
panded forms of/\x), f'\x), &c., and then perform the differenti- 
ations, remembering that dz is constant. As an example, we will 
change the independent variable from x to ^, where x^e" in the 

equation 

x^ d^y + 2xdy dx -^-y dx^ = 0. 

This is equivalent to 

or to x^f" {x) + 2xf{x) +/{x) = 0, 

since x is independent variable. 

Then, since a? = e*, dx = e'dz, 

"-^ ^""^'dx'edz' 

^///.N_^/W_ 1 .{dy\_ d?y-dydz 
J ^^~ dx ~ e'dz \edz) e'^dz' ' 

since z is independent variable, and therefore dz constant. 

Hence the equation becomes 

d'^y + dy dz ■'r y dz^ = 0, 



74 SUCCESSIVE DIFFERENTIATION. 

d^y dy 

where -p, -j^are the differential coefficients of y with respect to z.\ 

114. Having given an equation among the differentials of j; and 
1/ when X is independent* variable, to find the equivalent equation 
when^ is independent variable, or to change the independent variablel 
from a: to y. 

By dividing by some power of dx^ the given equation may be I 
transformed into that among the differential coefficients of y with 
respect to x. (Art. lO?)* 

Then, differentiating successively, remembering that y is inde- 
pendent variable, we obtain from the equation 

y S{(fxY-cPxdx 

&c. = &c. ; 
and by substituting these values in the original equation the required 
transformation is effected. 

The resulting equation may be transformed into that among the 
differential coefficients of ^ with respect to^, by dividing by some 
power of <Z^. (Art. lO?)* 

115. The methods of effecting these transformations will be easily 
remembered, if it is observed that 

(1) An equation with x as independent variable is immediately 
transformed into an equation involving only differential coefficients 
with respect to x, by division by some power of dx. 

(2) The expressions for the differential coefficients when any 
variable is made independent, are obtained from the expressions in 
which the independent variable is general, viz. 



/ 



«-l'/"w=i<S)'/"«=iM^<g}'^^- 



SUCCESSIVE DIFFERENTIATION. 7o 

by making the differential of the variable which is to be indepen- 
dent, constant, and performing the operations indicated, on that 
supposition. 

(3) If the variable selected for independent variable is neither 
X nor y, but a given function of x, dx must be expressed in terms of 
the differential of this quantity, and substituted in the above ex- 
pressions, and the operations performed on the supposition that this 
differential is constant. In like manner, if the new independent 
variable is given as a function of y, dy must be eliminated from 
the above expressions before performing the differentiations. 

116. We will take as an illustration the equation 
y = e^ cos X =f(x), 
and deduce from it equations among the differentials in various 
forms, and then apply the foregoing methods to obtain them one 
from another. By differentiating 

y = e"" cos X, 
we have, when the independent variable is general, 
dy - e^ (cos x - sin x) dx^ 
d^y = — 2e* sin xdx' + e'' (cos x — sin x) d^x. 
From these equations different relations among the differentials may 
be found: we will take that one from which cosx and sin a: dis- 
appear. Then, eliminating sin a: and cos a:, we obtain 

d'ydx - dy (^dx" + d^x) + 2y Jx^ = .............. (1). 

If we had differentiated with x as independent variable, and then 
eliminated sin x and cos x, we should have found 

d^y-^dydx-¥^ydx^=^0 (2). 

Again, the successive differential coefficients of ^ are 
f ( ar) = e' (cos x — sin x), 
f"{x) = -2e'dnx, 
whence we have, by eliminating sin x and cos Xy 

/'W-2/(^) + 2/(a:) = (3). 

Suppose now x = log^, the original equation becomes 
y = z cos (log z), 
whence, if z is independent variable, 

dy - {cos (log z) - sin (log 2)} dz, 

d'y = -~ {sin (log 2) + cos (log z)} dz^ ; 



76 SUCCESSIVE DIFFERENTIATION, 

whence we obtain 

z^cPy-zdydz + ^ydz'^^O (4). 

Lastly, differentiating the original equation with y for inde- 
pendent variable, we obtain 

dy = (^ (cos X - sin x) dx, 
= - 2e^ sin xdx^ + e* (cos x — sin ac) d?x\ 
whence, by eliminating sin x and cos x, we have 

d^xdy + 2d3i^dy-2ydx^ = (5). 

These five equations are all equivalent and may be deduced one 
from another by our previous rules. 

Thus (2) is obtained immediately from (l) by putting d^x = Oj 
and (3) from (2) by dividing by dx\ (Arts. 107, 108). 

(1) is deduced from (2) by the method of Art. IO9. Thus, first 
change (2) into (3), and then we have 



/'Vr^- ^ ^/^M - d'ydx-d'xdy 

^ ^'^^-Tx'^KTx)- d? • 



Substituting these values in (3), we have 
rf^y dx — d^xdu ^ du 

.-. d^ydx - dy (2dx^ + d^x) + 2ydx^ = 0, 

which is equation (1). 

Again, (1) may be transformed to (5) by making d'^y = 0, and to 
(4) by the method of Art. 110. 

To interchange (2) and (4), that is, to change the independent 
variable from x to z when x = log z, we must proceed as in Art. 112 
or 113. 

By dividing by dx^, (2) becomes identical with (3), and when ^ 
is independent variable, we have 

and since .r = log ^, dx = — , 



SUCCESSIVE DIFFERENTIATION. 77 

= -j-zd{zdy) since dz is constant, 
%{zd^xf -vdzdy) 

then, substituting in (S), we obtain 

z{zd'y^dzdij) ^^dy 

.'. z^d^y - Sft^^-^/y + ^ydz^ - 0, 
which is equation (4). 

In the same way (2) may be obtained from (4). 

To interchange (2) and (5), or to change the independent variable 
from X to yt we must proceed as in Art. 114. 

First reduce (2) to (3) as above; then 



f" [x) - -^ d[-j-\ ^ dy being now constant, 
dy d^x 



.•.rw=- 



dx'^ 



Then, substituting ijn (3), we have 

_dj^_^l__^dy a 

dx' ^dx^^y-^' 

or ^xdy + 9.dydx^ -9.yd3(^ = 0, 
which is equation (5). 

In the same way (2) may be obtained from (5). 

The student should work out those transformations which are 
not given at length in the text. 

For examples in the change of independent variables, see Gre- 
gory's Examples, Chap. iii. Sect. 1. 



CHAPTER VI. 

DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES 
AND IMPLICIT FUNCTIONS. 

117. The quantities we have hitherto differentiated, have always 
been given as explicit functions of a single variable. A function of 
one independent variable may, however, be expressed by an equa- 
tion of the form ti = F (x, y), where x and y are themselves connected 
by some other equation (Art. 8). In such a case the differential of 
u may be found by expressing one of the variables x and ^ as a 
function of the other by means of the given relation between them, 
whereby u will be reduced to the form of an explicit function of a 
single variable and may be differentiated by our former methods. 
Thus suppose we have 

where x and y are connected by the additional equation 

y = mx + c. 
By substituting this value of ^ in the former equation we obtain 
u = Jx'^ + {mx + cfy 
which may be differentiated by the rules of Chap. m. 

The following theorem will enable us to differentiate such funCr 
tions more readily, without the necessity of any previous elimination. 

118. Let ^ = F(x,y), (1), 

where x and y are connected by the equation 

^=/W -(a)- 

When w receives an increment Ax, y and u will receive definite 
corresponding increments Ay, Am which vanish with Ax, Au is 
given by the equation 
Au = F{x + Ax, y + A^) - F {x, y) 

= {F{x + Ax,y + Ay) -F(x + Ax, y)} + {F{x-¥ Ax, y) -F(x, y)]. 

The latter part of this expression is the increment which F (j*, y) 
would have received, if x alone had been variable and y constant ; 
it may be written A^F{x, y). 

The former part is the increment which F(x + Ax, y) would 
have received if y alone had been variable and x constant, and may 
in like manner be written AyF(x + Ax, y). 



DIFFERENTIATION, &C. 79 

Hence the above equation becomes, on dividing by one of the 
increments, as A:r, 

Am _ A^ F (j:, y) A^ F (x + Ax, i/) 
Ar ~ Ao: Ar ' 

which being true always, is true in the limit when the increments 
approach zero, 

X being written for x + Ao; in the second term because x and x + Ax 
become identical in the limit. 

Au dii 

Ax dx' 



Now It^^o 



Also U.^_.^^l^J-^l^^F'i.), 

where d^F (x, y) and F\x) signify the differential and differential 
coefficient respectively of F (x, y) obtained on the hypothesis that y- 
is constant and F {x, y) a function of x alone. Similarly 

A^F{x,y) _ d,F{x,y) 
"^=° Ay ~ dy ~ ^'^^' 

where dyF(x,y) and F'(y) are the differential and differential co- 
efficient respectively of F(x, y) obtained on the hypothesis that x is 
constant and F{xy y) a function of ^ alone; 

^ dyF {x, y) dy^^Y(v)^ 
dy dx ^'^^ dx 

d,F{x,y-) 
dx 

Substituting these values in the above equation and multiplying 
by dx, we obtain 

du = d^F{x,y)+d,F(x,y).. ...(3), 

or du = F{x)dx + F'(y)dy (4). 

F'(x), F'{y), being differential coefficients obtained by treating 
F{x, y) as a function of one variable only, ioaay be found by the 
methods of Chap, iii., and equation (4) will then give one relation 
between the three differentials du, dx and dy. 



80 DIFFERENTIATION OF 

By differentiating equation (2) we obtain 

by which equation together with (4), rlu may be expressed in terms 
either of dx or dy. 

d^F{x,y) and dyF{x,y) are sometimes written for conciseness 
dji dyU*, and are called the partial differentials of u with respect to 
a: andy respectively; F'(x), F'{y) are called the partial differential 
coefficients of u with respect to x and y respectively ; du is what we 
have already defined (in Art. 17), as the differential of m: to dis- 
tinguish it from the partial differentials, it may be termed the total 
differential of u. It is evidently the same quantity which we should 
have obtained by eliminating one of the variables between equations 
(l) and (2) and differentiating the result. 

The theorem proved above may therefore be stated thus : 
The total differential of a function of two variables, which are 
themselves functions of one another, equals the sum of the partial 
differentials with respect to the separate variables, or (with the ab- 
breviated notation) 

du = d^u 4- dyU. 

119, We will apply the theorem to the example of Art. 11 7. 

u = Jx^Tf, 
where y = mx + c. 

• In using this notation, it must be borne in mind that u is merely an abbrevia- 
tion for the function F (x, y). The pair of equations (1) (2) may be transformed into 
a number of diiFerent pairs, each of which would give the same value of m in terms of 
either of the other variables, but all leading to diiFerent values of dxu dyU. Thus 
the pair of equations u = xy"^ and j/ = e" is equivalent to the pair u = xye' and y = e* ; 
but the former give d^u = yHx = e^^'dx^ and dyU = 2xydy = 2x€'dy, while the latter 
give dxu = {l + x)ye''dx =(l + x)e^''dx and dyu^x^'dy. Hence when dxU and dyU 
occur, u must be understood to stand for the same function of x and y in both. 

Another variation of notation must also be noticed. The partial differential co- 
efficients F'{x) F\y) which with the notation of the text equal -^, -^- respectively, 

du duL 
are sometimes written — , — , the subscripts being omitted. These latter quantities 
dOG CLy 

then cease to be ratios, and become mere symbols equivalent to F'(x), F'{y), and the 

partial differentials are then written -;- , dx, -f- . dy. This is the notation used in 

dx dy 

Gregory's examples. We shall recur to the general subject of notation in Art. 138, 

infra. 



FUNCTIONS OP SEVERAL VARIABLES. 81 

„ , xdv ydy 

Here dji = . , du = - j ; 

Jx'+f J^' + f 

J , , X dx + ydtJ 

.'. du = dji + cLw = — , '^ • . 

If we wish to obtain -j- as a function of x we must employ the 

second equation to eliminate y and di/. Thus differentiating, we have 

dy = mdxj 

and substituting in the value of du for y and dy their values, we find 

, X dx + (mx + c)m dx 
du= . ^ 1 , 

Jx^ + {mx + cf 

du (1 + m^\ X + mc 

^•^ ' Jx' + {mx + cf ' 
the same result which we should have obtained by pursuing the 
method indicated in Art. IIT^ 

In like manner -— may be obtained. 
dy ^ 

120. The relation between x and^ was expressed in Art. 118., 
for simplicity's sake, by the explicit equation y=f{x); but the 
theorem du = dji + d^ju^ is independent of that assumption (which 
nowhere enters intd the proof) and is equally true whatever be the 
form of the relation between x and y provided they are in fact 
functions of one another. Thus the relation may be expressed by 
the implicit equation /(x, y) = 0, or by the pair of equations 

or more generally still by a system of n equations involving x,y and 
w — 1 other variables, since by solution or elimination any of these 
forms will give ^ as a function of x. 

121. As an example, let it be required to express du in terms 
of dz from the equations 



= ^in\^,x = e% y = z\ 

,« = cos(g. 

, fx\ - xdy 



Here dji = cos ( - ) — 



H. D. C. 



DIFFERENTIATION OP 



dx 


= e* dz, 






dy. 


= 2z dz. 






dU: 


= d^u + d^u. 








""©{'"; 


"*}, 






-©{— 


-^.e^zdz 

z' 


}. 






If we had first substituted for x and y their values, we should 
have had 



^ = sin(y. 



by differentiating which equation we may obtain the same result 
as before. 

In like manner, du may be expressed in terms of dx or dy. 

122. If there are more than two variables under the functional 
sign, it may be proved in a precisely similar manner that the total 
differential equals the sum of the partial differentials with respect to 
these variables. Thus, if u =f{x, y^ z ... v) where x^ y, z .,, v are 
connected by a sufficient number of equations to make u a function 
of one independent variable only*, 

du = d^ti + dyU 4- d^u . . . + d^iu 
Differentiation of Implicit Functions of one Independent Variable. 

123. We are now able to find -^ where x and y are connected 

by an implicit equation, without solving it with respect to either of 
them. Let x and y be connected by the equation 

F{x,y) = 0. 

Then if A^, Ay be corresponding increments of x and y^ the 
equation must be satisfied hy x + I\x and y + Ay. 

,', F(x + Ax,y + Ay) = 0, 

• This condition is essential. The very first step in Art. 118 depends upon it, 
since without it, an increment Ax of w would not produce corresponding increments 
of y and u. 



FUNCTIONS OF SEVERAL VARIABLES. 83 

Ax ~ Ax 

which being true always is true in the limit; 

... ^ZM = o, ordFix,y)^0. 

But dF(x, tf) is the sum of the partial differentials o£F(x, y), 
.-. F\x)djc^F'{ii)dy = 0, 

124. In like manner, if 

F{x, y, %) = 0, 
where x, y and z are connected by a second equation so as to make 
them functions of one independent variable the variables will have 
corresponding increments Ax, Ay, Az, all of which vanish together. 
Then 

F(x + Axj y + Ay, z + Az) = 0, 

AF{x, y, z) ^ F(x + Ax,y + Ay, s + Az)- F(x,y, z) ^ ^ 
"Ax Ax 

.-. ^^'^''jf' ""^ = 0, and dF(x, y,z) = 0; 

and generally if 

F(x,y,z...v) = 0, 
where the variables are connected by a sufficient number of equa- 
tions to make them functions of one independent variable, 

dF(x,y...v)==0, 
and therefore (Art. 122) 

F\x)dx -^-r(y)dy+,.. F'{v) dv = 0. 

125. Hence if n variables x^, x^.. .oc„ are connected by the w- 1 

equations 

jP,(ari, a?2...^„) = 0, 

Fs{xy,x^...x^ = 0, 



dx 

^'{x) dx + F'{y) dy + F'{z) dz = 0, 



F„_i(xi,a;2...x^) = 0, 
in which case they are functions of one independent variable only. 
We shall have 

F/ (x,) dx, + F/ (t,) dx,,.. + F/ (xj dx, = 0, 

6—2 



S-t DIFFERENTIATION OF 

F,' (or.) dx, + F,' {x,) dx,..,+ F,' (^„) dx„ = 0, 

FLi (^i) dx, + F'„_, (x,) dx,... + FL, W dx^ = 0. 

From these n — \ equations, any w — 2 of the differentials may be 
eliminated, leaving an equation to determine the ratio of the remain- 
ing two, that is, the differential coefficient of one of the corresponding 
variables with respect to the other. 

126. Let the explicit equations corresponding to 

^(^,.y) = o (1), 

be y=f{x\ x = <p{y) (2). 

Then since the forms of / and <p depend on that of F, those of/' and 
<p' must be connected with those of F'{x) and F'{y). The relations 
between them may be obtained as follows. 

The ratio of the differentials dx and di/ must be the same whether 
obtained from (l) or (2). But from (l) we have (Art. 123) 

F'(x)dx + F'(y)dy = 0. 
Also from (2) dy =f {x) dx, 

and dx = <p\i/) dy. 

These equations must be identical ; 

•• F'(s) ^^^ ^'(^)- 

127. Again, let three variables be connected by the two 
equations 

F(x,y,u) = (1), 

F,{x,y,u) = (2), 

and let the explicit equations obtained by the solution of (1) be 

^=f{^>2/)> x = <t>(u,y), y = yjr{x,u) (3). 

Then the ratios of the differentials may be determined by combining 
the equation dFi = with the differential of either of the above 
forms of equation (1), that is, by combining dF^ - with either of 
the following equations ; 

F'ix) dx + F'{y)dy + F'{u) du^O (4), 

du=fix)dx+f(y)dy (5), 

dx = <p\y)dy^4>'{u)du (6), 

dy = \lr'(x)dx + \f/\u)du (7), 

and since the ratios of the differentials must be the same from which 
ever form they are derived, the above equations must be identical. 



FUNCTIONS OP SEVERAL VARIABLES. 85 

Hence by dividing (4) by F' (ii) and comparing the coefficients 
with those of (5), yve obtain 

'^^^■^ F\uy ^^y^~ F'(m) 
So, from (4) and (6) we find in like manner. 

And from (4) and (7) 



,,„._^). ,,.,=-™ 



128. The same results may be obtained somewhat differently, 
as follows. 

If we differentiate equation (1) on the hypothesis that y is con- 
stant, we obtain an equation between the differentials of x and u^ 
which may be written in either of the forms 

or F'{x) d^x + -F' (w) du = 0, 
the former expressing the fact that y is constant, by assuming u to 
be expressed as a function of x and y^ and differentiated with respect 
to X alone, the latter by assuming x to be expressed as a function of 
u and yy and diffef entiated with respect to u alone. The one form 
tacitly refers to the first, the other to the second of equations (3). 
Again, by differentiating equations (3) partially, we have 
d^u =/' {x) dx, djc = 0' {iC) duy 
which being substituted, in the above give 

jF"(*) + F'(«)/(«) = 0, 



Similarly, by differentiating equation (1) on the hypotheses that 
X is constant and that u is constant respectively, we should have 
obtained 

F\u) J ^^' i^'(_uy 



the same relations as those found in the last article. 



86 DIFFERENTIATION OF 

129. Examples of the differentiation of implicit functions. 

1. To find -~- from the equation 

{x+yf = x^ +'my^. 
The equation may be put into the form 
u = {x ■¥ yY - x^ — my^ = 0; 
.*. dji = 2(x +y) dx — 2xdx = 2y dx, 
and dyU = 2(x + y)dy — 2my dy = 2{x + {1 — in)y} dy ; 
.*. = djt + dyU = 2ydx + 2{x+(l- m)y] dy, 

" dx .r+ (1 — rn)y ' 

The same result may be obtained by solving the equation with 
respect to y and differentiating the result, when the equations of 
Art. 126, will be verified. 

In differentiating an implicit function it is not necessary to re- 
duce it to the form u - 0. For if we have 

Ax,y) = <p{x,y), 
we obtain by equating the differentials of each side, 

f\x) dx +/ {y) dy = (p' {x) dx + <l>' (y) dy, 
the same equation which results from the difilgrentation of 

since d^u = {f'{x) — <p'{x)} dx, 
d,u^{f^(y)-<l>'(y)}dy. 
Thus, if we had differentiated the equation 
(x+yY^x^' + mf, 
as it stands, we should have got the equation 

2(x + y) (dx + dy) = 2xdx + 2my dy, 
which is identical with that found above. 

2. Let the three variables x, y, z be connected by the equations 

^2 yS ^ 

^■^P'*'^"^' •• (^)' 

x^ +/ + z' = r^ (2). 

Then from (1) 

2x dx 2y dy 2z dz ^ 



FUNCTIONS OP SEVERAL VARIABLES. 87 

and from (2) 2x dx -h 2i/ di/ + 2z dz = ; 

.*. eliminating dz, 

dj/ _ a^ c^ a: 

' ' dx }_ _}l y 

1_1 
Similarly, -=-^—^=|, 

a'- c' 

1 _i 

dz i_ i_ X 
a" b^ 

The same results may be obtained by first eliminating one of the 
variables between (1) and (2) and the relations of Art. 127, may be 
verified by solving equation (l) with respect to the several variables, 
and comparing the results of differentiating equation (1) and the 
explicit equations so obtained. 

130. In Art. 118, and in the other propositions of this chapter 
deduced from it, the quantities differentiated are functions of one 
independent variable only, for the number of variables in no case 
exceeds the number of equations by more than one. Thus, in the 
general case of Art. 125, we might have eliminated from the given 
equations, all but two of the variables, leaving an equation of the 
form 

from which, by the fundamental definition we could have obtained 
dx^ =f'(x^dx2. 

The differential equations of Art. 125, merely afford a different 
and generally a readier method of determining the ratio of the same 
differentials. 

If however the number of equations between the variables is 
not sufficient to make them functions of a single independent variable,, 
the reasoning of Art. 118, not merely fails, but the result becomes 
unintelligible, because the definition on which it is based ceases to 
have any meaning. 



88 DIFFERENTIATION OF 

Thus let u, X, y be connected by a single equation whose implicit 
and explicit forms are 

F{u,x,y):=^0, (1), 

'^^=f{p,y\ (2). 

x = <^{u,y\ (3), 

y = yl^{u, x), (4). 

Then it will be found impossible to attach any meaning to the 
differentials du, dx, dy. 

For the only definition yet given of a differential is that of 
Art. 17, which is in terms restricted to the case of functions of one 
independent variable^ and may be stated as follows. 

If X and y are functions of one another and an increment Aj; be 
given to x, y will receive a corresponding increment Ay bearing a 
definite ratio to Aj; and vanishing with it. Such increments will 
have a limiting ratio, and this limiting ratio is denoted by the 

ratio -^ . 
dx 

For the sake of perspicuity we have, in Art. 118, prefixed the 
term total to the differential of m, but we added nothing to the 

definition and the -r- of that article means nothing more than the 

A 7/: 

limiting value of -^ which might have been obtained by first elimi- 
i^x 

nating y between the given equations. So the partial differentials 

of Art. 118 are merely the differentials of m, obtained from the 

fundamental definition on a particular hypothesis, and involve no 

extension of the definition. 

Now if we endeavour to apply the definition to an equation such 
as (1) or (2), we find that it gives no meaning to the term differ- 
ential {i. e. total differential) of m, because the hypothesis on which 
the definition rests, viz., that m is a function of some one other variable 
is no longer true. 

Thus, if 

X + Aj;, y + A^, u + A?/, 

be values of the variables which satisfy equation (2) we have 

Am =f{x + A^, y + Ay) -f{x, y), 

and no other relation exist between the increments. When an arbi- 



FUNCTIONS OF SEVERAL VARIABLES. 89 

trary value is given to Aj:, Au does not assume a definite value, 

until Ay also has been arbitrarily determined. Hence Am does not 

bear a definite ratio either to Ax or Ai/ but depends on both of them, 

and no such thing exists as a definite limiting value of either of the 

Am Am 
ratios -^— , -r— , on which our definition of a differential was founded. 
Ax Ay' 

In short, the definition of Art. 17, was framed exclusively with re- 
ference to functions of one independent variable and has no applica- 
tion to the case we are now considering. 

It is otherwise however with respect to the partial differentials 
and differential coefficients. The values of /'(a?), f{y) may be ob- 
tained as before, since the supposition of only one variable receiving 
an increment, while the other remains constant, reduces the function, 
so far as these operations are concerned, to a function of one variable 
only, and brings it within the scope of the original definition. Thus, 
if X receives an increment Ax while y remains constant, u will re- 
ceive a corresponding increment A^m, bearing a definite ratio to Ax. 

-~— will therefore have a limiting value when Ax approaches zero, 

and we may define the differentials d^u arid dx to be any quantities 
whose ratio equals that limiting ratio, whence we have (as before), 

Hence, partial differentials of functions of two independent vari-, 
ables have the same meaning as when the variables are connected. 
Total differentials of such functions have no meaning whatever. 

If then, the term total differential is to be applied at all to func- 
tions of two independent variables, it must be by virtue of some new 
definition. As yet, it is undefined, and we are at liberty either to 
leave it undefined and unused, or to define it in any way which may 
be found convenient. In selecting a definition, we shall seek one, 
which, while it applies to functions of two independent variables, in- 
cludes within it the definition before given of a total differential of 
a function of one independent variable only. The advantage of such 
a definition is obvious as it will enable us to work with differentials, 
without enquiring in each case into the number of independent vari- 

• We cannot now use U^^^ instead of ^^^^.-^ because Ad? and ^y do not necessarily 
vanish together. 



90 DIFFERENTIATION OF 

ables. Indeed, unless we attached to the term when extended to 
functions of two independent variables, a meaning analogous to that 
which it bears when there is only one, it would be better to leave it 
undefined and to confine its use exclusively to the latter case. 

In order to maintain this analogy we define the total differential 
as follows. 

Def. The total difl^erential of a function of two independent 
variables is the sum of the partial differentials. 

When expressed in analytical language (with reference to 
equation 2,), the definition is as follows. The total differential, du, 
of u is a quantity which satisfies the equation 
du - d^u + dyUj 
or du^f(x)dx^f(y)dy (5). 

So {with reference to equations (S) and (4)} the total differentials 
of X and y are defined by the equations 

dx = (p\u)du + <p'{y)dy, (6), 

and dy = yl^'{u)du + ylr\y)dy, (7), 

or still more generally with reference to the implicit equation (1), 
the definition assumes the following shape. 

If dx, dy, du are any three quantities which satisfy the equation 

F'{x)da: + F'{y)dy + F'{z)dz=0, (8), 

dx, dy and du are called the total differentials of x, y and u. 

The several equations (5) (6) (7) and (8) are all identical, by 
virtue of the necessary relations which exist between the forms of 
F,f,<p and yp^, as appears from Art. 127^ so that the differentials on 
the right hand side of equations (5) (6) and (7) as well as those on 
the left, represent total differentials. 

When w is a function of more than two independent variables, 
the total differential is defined in like manner as the sum of the 
partial differentials. Thus, if 

F{x,y ...u,v} = 0, 
one of the explicit equations being 

the total differentials of the variables are defined by the equation 
F'{x) dx + F\y) dy ,,.^F' (w) du + F'{y) dv = 0, 



FUNCTIONS OF SEVERAL VARIABLES. 91 

or by any of the equivalent equation s^ as 

du=^f{x)dx+f\y)dy+ ...f{v)dv. 

These equations, therefore, having been deduced from the original 
definition of Art. 17j in the only case where that definition applies, 
and being themselves taken as the definition in all other cases, are 
universally true whatever may be the number of independent 
variables. 

131. Hence, generally, if n variables are connected by r equa- 
tions, 

i^i(a:i,cr3 ... ^„) = 0, 



i^,(a:,,^2...^„)=0, 
the relations among the total differentials will in all cases be given by 
the equations 



total differentials wil 

IS 

F\{x^) dx^ + F\{x^ dx^...-\- F\{x„) dx„ = 0, 
-F'a(a:,) dxi + F\{x^) dx^ . . . + F^{x,)dx^ = 0, 

F'J^x,) dx, + r^x^ dx^,,.+ F\(x„) dx„ = 0. 

132. Whatever be the number of equations between the varia- 
bles, there will be the same number between their differentials. 
Hence, when the variables are functions of one independent variable, 
the number of equations between the differentials will be one less 
than the number of differentials, and one differential may have an 
arbitrary value given to it when the others will take definite corre- 
sponding values. When there are two independent variables, arbi- 
trary values may be given to any two of the differentials, and in 
general as many differentials may be arbitrarily determined as there 
are independent variables in the original equations. 

Thus when the only equation between the variables is 
Fix, y, u) = 0, 

dx, dy, du are by the definition any quantities which satisfy the 
equation 

F'{x)dx + F'iy)dy + F\u)du = 0, 
To any two of them, therefore, we may give what values we 
please, when the above equation determines the value of the other. 
So in all cases where there is but one equation given between the 



92 DIFFERENTIATION OF 

variables, all but one of the differentials may have arbitrary values 
assigned to them. 

133. The definitions of Art. 130 will not at once enable the 
student to attach any other idea to differentials of functions of several 
independent variables than that of quantities which satisfy a par- 
ticular equation. There is nothing in this notion analogous to the 
fundamental idea of differentials of a function of one independent 
variable, viz. quantities whose ratio equals the limiting ratio of the 
increments. Such an idea is however involved in the definition of 
Art. 130, and it is important to consider it, on account of its value in 
geometrical and other applications of the subject. 

An alteration in the form of Art 17 will help to make this clear. 
JjQty=f{x) be any equation connecting x and y ; Ax, Ay, corre- 
sponding increments of x and y. 

Conceive two quantities Ixy ly which do not vanish with Aj? and 
Ay but which (as Ax and Ay vary) so change that ly always bears 
the same ratio to Ix which Ay bears to Ax. Then, instead of the 
definition of Art. 17j we might have defined the differentials to be 
the limiting values of Ix and By. For this would have given us 

The definition so modified may readily be extended to functions 
of any number of independent variables when it will be expressed 
as follows. 

Def. The total differentials of variables connected by any num- 
ber of equations, are the limiting values of any quantities bearing the 
same proportion to one another as the increments, but not vanishing 
with them. 

It remains to shew that this is identical with the analytical defini- 
tion of Art. 130, which may be done as follows. 
Let u =fi^, y). 

Let Ax, Ay and Am be any increments consistent with the above 
equation. Then 

Am =/(x + Ax, y + Ay) -/(x, y) 

= {JX^ + ^^,^ + %) -A^ + A^, !/)} + {/{^ + ^^^ y) -/(-r, y)} 
= ^Ji.^ + A-^, y) + Ax/(.r, y) 
= P.Ax + Q.Ay, 



FUNCTIONS OF SEVERAL VARIABLES. 93 

Ax Ay 

Therefore if Bw, Bar, ly be any quantities bearing the same pro- 
portion to one another as Aw, Ax, Ay, we have 
lu:=¥.lx^Qi.ly, 
In the limits when all the increments approach zero, Bm, Ix and 
ly become, according to the above definition, du^ dx and dy, and we 
have 

du = UA=oP.dx + U^^Q.dy. 

Also as in Art. 118, 

and lt,^Q = lU^^Sl^^^lA =.fXy) ; 

.-. du=f{x)dx+f{y)dy, 
which is the equation by which the differentials were defined in Art. 
130. A similar proof may be applied to functions of any number of 
variables *. 

• The mode in which the definition of a differential is extended in the text to 
cases not contemplated by the original definition may occasion some little difficulty. 
The principle however is not peculiar to this subject and is precisely the same as that 
adopted with reference to the fundamental definitions of Algebra. One instance will 
be familiar to most readers : «*" is defined when wi is a positive integer as the product 
obtained by multiplying a by itself m - 1 times. Hence result the equations 

a^.a" -«"•+", 
«•" 
o» ' 

(a™)" = a""", 
trhe above definition is obviously unmeaning when m is negative or fractional. "We 
can attach no idea for instance to the multiplication of a by itself half a time. A new 
definition becomes necessary, and accordingly we define a" (when m is other than a 
positive integer) to be such a quantity as satisfies the above equations, which then 
hold universally. This is precisely analogous to what we did in Art 130 of the text. 
Having so defined a"* in all cases, the next question is whether we can attach any 
meaning to it beyond that of a quantity satisfying particular equations, and the 

result obtained is that a-™ means — and a" means ^a. The analogy between this 

explication of the definition of a™ and the process of Art. 133 will be readily seen. 

The primary rules of algebra require to be extended beyond their original 
meaning in the same way as the definition of a*^, although such an extension is com- 
monly omitted in elementary works for the sake of evading a difficulty at the 



94 



DIFFERENTIATION OF 



134. It may be observed, that to any two of the increments 
Aw, Ar, A^, in the last Article, we may give arbitrary values. 
Hence two of the quantities Sm, Zx, ly, which are in the same pro- 
portion, will also be arbitrary, and the same must be true of their 
limiting values du, dx, dy^ a. conclusion somewhat differently arrived 
at in Art. 132. 



135. The following figure will be intelligible to those who are 
acquainted with Analytical Geometry of three dimensions, and will 
shew the geometrical meaning of the differentials of functions of two 
independent variables. 




PQRS represents a portion of the surface whose equation is 
2 =/(j?, y)} the axis of a: being parallel to Pm, that of ^ parallel to Pn, 
and that of z downwards (which makes the figure clearer than if 
z were measured, as usual, upwards). 
Let P be the point x, y, z; 

PM=Ar, PN=Ay. 
Then the increment of z corresponding to an increment of a; alone 

threshold of the subject, and the rules are in fact proved only for particular cases and 
then tacitly assumed to be universally true. In strictness they can only be gene- 
ralized by a method similar to that of the text and affording a useful illustration of it. 
On this subject the reader is referred to Ouui's Mathematical Analysis, trans- 
lated by ElUs, liondon, 1843. 



FUNCTIONS OP SEVERAL VARIABLES. 



95 



is MQ, that corresponding to an increment of f/ alone is NR^ and 
that corresponding to increments of both variables is LS ; 

Also RV and QU being drawn parallel to x and i/, 

VS = A,/(a;, y + A^), VS = d.J{x + A^, y\ 

since F^S' is what MQ becomes when ^ + A^ is put in the place of 
yj and US the value of 'NR when x + Aa: is put for x. 

If Tm, Pn are lines o/^ aw^ length parallel to x and y respectively, 
Pqsr the tangent plane at P, and therefore Pq, Pr the tangents to 
the sections PQ, PRj we may put 

Pm = c?.r, Pn — dy ; 

whence we have (as in Art. 19) 

mq = d^z = lu, nr = d^z = Iv. 
Also since qs is parallel to Pr, us = nr; 

.'. Is =^lu + us = lu + nr=- d^z + r/^z, 
.-. ls = dz. 

Hence the significance of the definition of Art. 133 appears. For 
as PM and PiNT diminish, the surface becomes more and more nearly 
identical with its tangent plane ; and since the increments are the 
co-ordinates from P of any point whatever {S) of the surface, the 
differentials (or me limiting values of quantities in the same pro- 
portion as the increments) must be the co-ordinates from P of any 
point whatever {s) of the tangent plane at P; which agrees with 
the above construction. 

136. Exactly as in the case of equations of one independent 
variable, so when there are more, those variables whose differentials 
are assumed to be constant are called the independent variables, and 
when none of the differentials are considered constant, the inde- 
pendent variables are said to be general. 

Since the partial differentials and differential coefficients are gene- 
rally variable, we may have second, third ...n*^ partial differential 
coefficients and partial differentials, as has already been seen with 
respect to total differentials of functions of one variable. The same 
will also be true of the total differentials of functions of several 
variables, these being generally variable. 



96 



DIFFERENTIATION OF 



Again, if u is a function of two variables x and y, the partial 
differential or differential coefficient of u with respect to x will 
generally be a function of ^ as well as x, and may therefore be dif- 
ferentiated partially with respect to y. Hence we may have such 
quantities as 



^.KW}, d^{dy{u)}, ~: 



dy ' dx 

These latter may be represented in the functional notation by the 
symbols f\y, x), f'{x, y)^ these symbols being understood to mean 
that we are to differentiate u twice, first partially with respect to Xy 
and then partially with respect to y, and in the reverse order re- 
spectively. We shall now shew that the order in which such dif- 
ferentiations are performed is immaterial. 



137. To shew that 



'-&} <£) 



dx dy 

By the definition of partial differentials, we have, if m = f{x^ y), 



dy 



i f{x,y + I^y)-f{x,y-) \ 



Now when x becomes x + ^x, the above expression becomes 
, i f{x + Aj?, ^ + Ay) -f{x + Aj:, y) \ 



d^u . 



and A^ -f— is the difference between these two values : 
dy 



<%) 



\ ^ , which by the definition = It^^^ — r—^ 
ax Aj: 



= "Aa^O 



It 



U / /(■r+Aar,.y+Ay)-X^+Ajr,y) \ f f{x,y+/^y)~f{x,y) \\ 

_ J 

\ f{x + ^x,y + Ay) -f(x + Ax, y) -/(x, y + Ax) +f(x, y) ) 
[ Ax Ay r 

writing It^^ because it appears by the previous step that the limits 
are to be taken by making both Ax and Ay approach zero. 



The value of 



dy 



may be found in the same way, and will 



FUNCTIONS OF SEVERAL VARIABLES. 97 

be precisely the same as that of , since the above expression 

(being perfectly symmetrical) remains unaltered when x and y are 
interchanged. 

ax dij 

and the symbol, /''(a:, y) may be used to denote either of them. 

Cor. 1. If or and y are independent variables, we obtain from 
the above equation, since dx^ dy are constant, 

d^dyii _ dydji 
dxdy dxdy' 

In this case therefore, d^d^u = d^dji. 

It must be remembered that this equation does not hold, unless 
.r and y are independent variables. 

, , , . ''-(f) o. . , 

Cor. 2. From the equation ^ = — 'i it is easily seen 

that when a function is to be differentiated m times partially with 
respect to one variable, and n times with respect to another, the 
order in which the differentiations are performed is immaterial ; for 
by repeatedly interchanging any two contiguous symbols (by virtue 
of the above theorem) the order may evidently be changed in any 
manner we please. 

138. On the notations employed to represent successive partial 
differentials and differential coefficients. 

Having employed djij d^u to represent the partial differentials 
of u with respect to x and y^ the successive partial differentials must, 
consistently with our notation in the case of functions of one inde- 
pendent variable, be represented by 

4w, d^^u (//)m, dyU, d^u d^^'^uy djyu rf,X"^ (1)» 

and representing the partial differential coefficients of w or F{x,y) by 
F'{x) and F\y)^ the successive partial differential coefficients with 
respect to x alone and y alone must be represented by 

FXai), F%x) Fi'^^x), FXy), F^^(y) F^-^iy) ... (2). 

When we first find the partial differential coefficient of u with 
respect to x, and then with respect to y^ the result has been repre- 

H. D. C. 7 



98 DIFFERENTIATION OP 

sented by F''{x, y) ; a notation which cannot conveniently be ex- 
tended to the case where the function is differentiated m times with 
respect to x^ and n times with respect to y. This might be expressed 
by the symbol F^'^^^x^, y^, but such a notation would be very in- 
convenient in practice. 

Since, as far as partial differentiation is concerned, u may be con- 
sidered as a function of one variable only, the values of the differ- 
ential coefficients in terms of the differentials will be, (Art. 98), when 
the independent variables are general, 

^ i® &c ^ ^M &e -1® &c rs) 
dx' dx '^^' dy' dy '^^* T^ . &c...(3;, 

which differ from the expressions in the case of one independent 
variable, only by the use of the subscripts, to denote the variable 
with respect to which the differentiations are performed. When x 
and y are both independent variables, these become (Art. 99). 
dj£ d^u dj'u dyti dyU d^'u dJyU d^d^u . . 
dx' da^"'dx''' dy' dy^'" dy^ ' dxdy"'~dbe^'"^ ^' 

These notations (2), (3), (4) are precise, but the functional nota- 
tion (2) cannot be conveniently extended to the general case ; (3) if 
used merely as a notation to represent differential coefficients, is 
excessively cumbersome, and (4) holds only when x and y are made 
independent variables. To obviate these objections a less expressive 
but more convenient notation is generally employed. 

In the first place the expressions (4) are used for the differential 
coefficients, not only when x and y are independent variables, but in 
all cases. When this is done, the numerators cease to represent the 
successive differentials as heretofore ; but this will occasion no error 
if we regard those expressions merely as symbols, and not as frac- 
tions. This inseparability of numerator and denominator is indi- 
cated by enclosing them in brackets ; and since the variable with 
respect to which the differentiations are performed is then sufficiently 
indicated by the inseparable denominator, the subscripts- also are 
omitted. 

The partial differential coefficients are then represented by 
/^N fdhC\ (^\ fdu\ (dhi\ fd^\ 
\dx)' \dxV"'\dxV' \dy)' \df)"\dyV' 

(d'u\ fd-^"u\ ' 
\dxdyJ"'\dx"'dy'^J'"^ ^' 



FUNCTIONS OF SEVERAL VARIABLES. 99 

Where the nature of the problem precludes any probability of 
error, even the brackets are frequently omitted. , 

The notation (2) is recommended in preference, where no differ- 
ential coefficients beyond the second occur ; in other cases (5) should 
be employed. The latter however is more generally adopted in all 
cases. For the sake of familiarising the reader with it, we shall 
employ it in the following propositions. 

139. To express the successive total differentials of u or F{x, y) 
in terms of those of x and y, and the partial differential coefficients of 
w, neither x nor y being independent variables. 

We have 
du = d^u + dyU (1), 

and d(|) =4(|) +<*„ (g) = (~)<l. + (g)rfy. 

Substituting these values, remembering thatf j =( ^ J, we 

have 

-•• (2) 

Equations (l) and (2) give du and d^u, and by proceeding in the 
same way the expressions for the differentials of higher orders may 
be obtained, but they increase rapidly in complexity as we advance, 
when X and y are not independent variables. 

140. The expressions for the successive total differentials in 
terms of the successive partial differentials of u are easily obtained 
Thus 

du = d^u + dyUj 

d^u — d^ [d^u + dyu] + dy {dji + dyu}^ 

.'. d^u = dfu + d^ dyU + dy d^u + d^u ; 

7—2 



100 



DIFFERENTIATION OF 



and similar equations for the differentials of higher orders. It will 
be observed that d^ dyU and d^ d^u are not equal because a; and 7/ are 
not independent variables. 

Equations in the above form are of little use^ but we may deduce 
from them the equations of the preceding Article, which we will 
now do as an illustration of the principles above explained. 

141. From the equation 

d^u = dfu + d^ dyU + dy dji + d^u^ 
to express dhi in terms of the partial differential coefficients of z^. 

similarly 

Adding these quantities, we obtain 

+ £) {djx + d^ dx) + {^ {d^dy + d^ dij\ 

since d^ dx + dydx = d^x and d^ dy + dydy = d^y. 

142. The expressions which we have found for the differentials 
of u^ hold not only when x and y are connected by some further 
equation^ when one at least of the differentials dx^ dy must be vari- 
able, but also when there is no other relation between the variables 
than the equation u = F{x, y), so long as the independent variables 
are left general. On account of the complexity of those expressions 
this is seldom done, but the variables under the functional sign are 
almost always made the independent variables. 



FUNCTIONS OF SEVERAL VARIABLES. 101 

143. To express the successive total differentials of u or Fix, y) 
in terms of the partial differential coefficients when x and y are 
independent variables. 

We have du = d^u + d^u^ 



•••*'=S)''^-^ (I) ''•?'• 



Also d^u = djiu + dyC 



since dx and dy are constant. 

By differentiating dSi, we obtain, in like manner, 

In these equations it may be observed that the coefficients and 

the indices of the operations -7-, -7- and of the differentials dx, dy 

are the same as those of the binomia theorem. By assuming this 
for d"U3 it may be proved for d"^^u, by differentiation, and being true 
for dJ^Uj d^u, is therefore generally true. This theorem may be 
written in an abbreviated form, thus, 

/ dy 

with the convention that wherever in the expansion we find [-7-) dx^, 
we are to write {jz^j^^^i and where we find f -^ j l-r-jdx^dx'^j to 

The theorem may easily be extended to functions of any number 
of variables. Thus if 

u = F{xi, X2...x^)j 

Xi, X2...x,n being independent variables. 

144. To change the independent variables in an equation in- 
volving partial differential coefficients of a function of two inde- 
pendent variables, from one system to another. 

Let u = F(x,y), 

and let it be required to change the independent variables from 



102 DIFFERENTIATION OP 

X and y io r and d, where these latter are connected with the former 
by the equations 

By substituting the values of x and y obtained from these equations, 
w becomes a function of r and d. Also 

\dxj ~ \dr) \dx) ^ \ddj \dx) ' 
f-pK { J-) being the differential coefficients of r and on the sup- 
position that y is constant, and therefore equal to ^'{pc\ "^'{x), which 
may be found from the above equations. 
In like manner, we have 



fdu\ _ fdu\ fdr\ (du\ fdd\ 
\dy) - \dr) \dy) ^ \dd) \dy) ' 



by substituting which values in the given equation the transforma- 
tion is effected. 

The expressions for the differential coefficients of higher orders 
are obtained in a similar manner, but the general expressions are 
somewhat complicated, and all that is necessary is to pursue the 
same method in any particular case. 

145. When the equations connecting r and 6 with x and y can- 
not be reduced to such a form as to give r and 6 explicitly in terms 
of X and y, we must proceed as follows. 

Let x = <p{r,e), y = yk{r,e). 

/du\ fdu\ fdx\ {du\ fdy\ 

Eliminating successively (-r-) ^^d (^-j, we obtain 

(dM\ (dy\ _ /^\ (dy\ 
\dr) \ddj \dej \dr) 



fdu\ 

\dxj-(dx\ fdy\_(d_x\ (dyy 

\dr) \dej \ddj \drj 

/du\ /dx\ /du\ rdx\ 

(du\ \dd) \d?) ~ \Tr) \dd) 

\dyj'-(dx\ fdy\_fd_x\ (dj\ 

\dr) \dd) \ddj \dr) 



FUNCTIONS OF SEVERAL VARIABLES. 103 

The partial differential coefficients of x and y are immediately de- 
ducible from the equations connecting x and y with r and 6, and the 
transformation is effected by substituting in the given equation the 

above values of f-^ j and \-t-j' 

146. We will take as an example the expression 
fdR\ fdR\ 

the new independent variables being determined by the equations 
r^ = ^'+/, tan0=^. 

X 



x^sec'd' 



fdR\ „ /'dR\ sin 

Also (f) = (f)Um^^„, 

\dyj \dr J r \ddj xsec^ 6 

/dR\ . ^ /dR\ cos 

-[w)''''^-'[-de)-T-' 

/dR\ fdR\ (dR\, . . _. 

fdR\ \ii cos - a: sin 1 fdR\ 
'U)^ r } = '"W- 



We will treat the same function by the method of Art. 145, em- 
ploying the equations 

x = r cos d, y = r sin Q, 

--- (f)=(f)--(f)-^' 



''— '*=-(S)'-^'"^+©'-'°^*' 



\ddj 

(S)=(f)---(S)^. 



r ' 

whence, as before, the expression is reduced to the form 

'dR' 



if) 



Examples wiU be found in Gregory's Examples, Chap. i. ii. 
and III. 



CHAPTER VII. 

DEVELOPMENT OF FUNCTIONS. 

147. To expand f{x + h) in ascending powers of h. 

Lemma. If (p {x) be any function of x, sucli that (p(a) = and 
f})'{x) is positive for all values of a; from a to a + k inclusive, then 
<l){x) will (between the same limits) have the same sign as A; and 
if <p' (x) be negative for all values of x within those limits, then <p (x) 
will have a different sign from h. 

This follows at once from the consideration, that the differential 
coefficient of (p {x) with respect to x equals the ratio of the rates of 
increase of <p {x) and x^. 

Let now F(x) be a function of x such that F(^a) = 0, and let A 



and B be the greatest and least values of — — M-^ 

71 \X ~~ CI J 

from aw^ioa + h. Then between these limits, 



when X varies 



A^ 



and 



n{x- a)"-' 



n(x - ay-' 
are constantly positive, 

.-. An{x- ay-' - F'{x) and F' {x) -Bn(x-- a)'^\ 

are both constantly positive or constantly negative, since (x — a)"~^ 
cannot change its sign within the given limits ; 

.-. if ^1 (x) = A{x-ay-F {x) and 0^ (x) = F(x)-B(x- ay, 

* The Lemma becomes more obvious from the followmg construction. Let OAB 
be the axis of x, OA being equal to a, and OB to a + h, h being in this case positive. 

Fig. 1. Fig. 2. 



r,' 






Then the curve y = 4,{x) must by the assumed conditions pass through A^ and (if the 
differential coefficient is positive from A to B) its form must be that of figure 1, and 
y will be positive between A and B. So if A is negative, and therefore OB' less than 
OA, y -will be negative. If ^' is negative, the curve will be asin figure 2. 



DEVELOPMENT OF FUNCTIONS. 105 

the above quantities will be the values of </)/(^) and (pzipc), and 
01 (x) and <^2 {^) will satisfy the conditions of the lemma, and will 
therefore both be positive or both negative within the given limits, 

.-. A - 7 — Mr^i and -. — Mr„ - By 

have the same sign ; wherefore, giving to x one of the admissible 

values, viz. a + hy — ^-j~ — - lies between ^ and B. 

This may be expressed by the equation 

where Q^ has some unknown value between and 1, since the 
quantity on the right of the equation may, by varying Q^ from to 1, 
be made to assume all values from B Xo Ay some one of which must, 
therefore satisfy the above equation. 

If not only F (a) but also F\a) equals zero, we may prove in the 
same way that 

F'(a + eji) _ F"{a + eji) 

where 6^ has some imknown value between and ^, , and therefore 
a fortiori between and 1. 

If all the quantities F(a), F'{ci) . . . jp("-') («)^ equal zero, we may 
obtain a series of 6quations similar to the above, of which the last is 

{On-^hy - 1 

where the value of 6 must lie between and 1. 

Collecting these equations, we have (with this last set of con- 
ditions), 

F(a + k) _F^"^{a + ek) . . 

hr'~ \n •••• ^ ^' 

where Q has some unknown value between and 1 . 

We will now give a particular value to F(^), viz. 

^(*)=/W- {/(«)+/(«) (^-«)+/"(«) ^g^-+/'"-"(«) ^^' } • 

By differentiating F {x) successively, we obtain. 



106 DEVELOPMENT OF FUNCTIONS. 

J"'W=/"W-{/"(«) +/"'(«)(^-°) + - +/'"""(") 7^=1^ }> 

F«(*) =/••>(*) - { /"(a) +/'«'(«) (X - a) + . . . +/"-"(«) ^^^S-i } ' 
F>-\x)=f'-\x)-{f-'(a)}, 

All the terms within the brackets in the above expressions, ex- 
cept the first, are multiplied by some power of {x-a), and will 
therefore vanish when x is made equal to a, unless some of their 
coefficients are infinite. 

If, therefore, f{x) be such a function that none of the differential 
coefficients /'(a) f"{a) ...f^\a) are infinite, we shall have 
F (a) = 0, F\a) = . . . J'W(a) = . . . F^\a) = 0, 
and F\x)=f{x), 

F(x) therefore satisfies the conditions necessary to equation (1) and 

by substituting the values of F(x) and F^^^x) in that equation it 

becomes 

/(« + h) =/{«) +/'(«) h +/" («) g ... +/<-') (a) ^ +/" (a + eh) ^, 

~ (2). 

In this expression we may give to n any value we please (if by 
so doing we introduce no infinite differential coefficients) ; and if 
none of the differential coefficients of /(a:) become infinite when 
x = a, n may be taken indefinitely large, and we have 

fia + h)=fia)+r{a)k+f^\a)^ + &c.inin/. (3). 

The series in this form is called Taylor's theorem. 

The last term in expression (2) is evidently the difference between 
this infinite series and its first (n) terms. Its value cannot be exactly 
determined, since 6 is unknown, but we can determine the greatest 
and least values which it can assume while 6 lies between and 1. 
Since the remainder of the series after (n) terms must lie between 
these values, they are called ike limits of ike remainder of Taylor s 
series, 

148. The above series is true for all values of h, and therefore 
among others for the value dx. If u =/(x) and k = dx, the general 



DEVELOPMENT OP FUNCTIONS. 107 

term of the series becomes -^^ — M — , or (if x is taken for indepen- 
dent variable) -r— . 

[r 

In this case, therefore, we have 

/{a; + ax) = 7i + du + j^ + . . . + -p- ...tn mf. 

149. As an example of the application of Taylor's theorem, let 
f{x) = log Xj and let it be required to expand log (1 + j:) in powers 
of or. 

We have 

f{x) = \ogx, .-. /(l) = log(l) = 0. 



0?* 



a;" 

.-. log(l + x)=x-|-H-[2g... + (-l)-[n^^....-«»jf. 

The limits of the remainder after n terms will be the greatest 

and least values of *^— ^ -x"" from 6 = to = 1, and 

\n 

[w n {\-\-QxY 
Hence the limits are (- 1)""' - and (- 1)"-' — 



n ^ ' w(l+.r)"* 

150. To expand f{x) in ascending powers of a;. 
If in the expansion of/(« + h) given by Taylor's theorem, we put 
a = 0, A = a;, we have, 

/(«) =/(o) +/'(o)* +/"(o) I - +/"-(o) ^+/"'(fe) ^; 

or f(x) =/(0) +/'(0)* +/"(0) ^+ . • •• » <■»/. 



108 DEVELOPMENT OF FUNCTIONS. 

This is called Stirling's or Maclaurin's theorem, and is, as we 
see, only a particular case of Taylor's theorem. 

The condition that none of the quantities f{0), f{0), &c. should 
be infinite is necessary to its truth in the second form, the first re- 
quiring only that those up to the rf" order should be finite. 

151. As an example of the application of Stirling's theorem, let 
it be required to expand sin x in powers of a:. 
We have f(x) = sin x, .-. /(O) = 0, 

fix) = cos X, .-. /(0) = 1, 

/'(^) = -sina:, -'> r(P) = 0, 



/"'W= 


sin (x + n 


i)' - 


/'"'(O) = (-!)*'" 


-'if«i 








- 


= if w is even. 


.*. sin a; = a? 


-i^ 




....+(- 


'' |2»i-l 


.,.in in 



The limits of the remainder after the term involving a^"'-'^ will be the 



X 



greatest and least values 0^/""^^]^' 

and f^\ex) 1^ = sin (0^ + »t^) g^ , 

= (-ir sin(^^) -^ — . 

x^^ 
Hence the limits are and (- 1)"' sin x j^. 

152. To determine the nature of the expansion of /(a + h) when 
Taylor's theorem fails. 

This will happen when any of the functions f{a),f{a),f\a), &c. 
become infinite. 

(1) Suppose f{a) = oo . 

Then f{x) can generally be expressed in the form 

-^^^■^-{x-ar' 
where m has such a value that (p{a) is finite. Hence Taylor's 
theorem holds with respect to </) (a + A), 

., fia + h) = ^-^1^ = h- {(/>(«) + cp' (a) h + rW i /^^ + &c.} ; 



DEVELOPMENT OF FUNCTIONST. 109 

2. e. the expansion of f{a + h) contains negative powers of h when 
f{a) = 03 . 

(2) Suppose that the first of the series of functions /(a;), f{x)y 
f"{x), &c. which becomes infinite when oc = a^ is f^''^{x). 

Let F{x) =/<x) - {/(a) ^f(a){x -a)... +f'-'\a) ^^^^ } . 

Then, since all the quantities introduced in the above expression are 
finite, we shall have, as before, (Art. 147), 

i^(a) = 0, F'(a) = 0, F^'-\a) = 0, F^\a) =fXa) = o:, . 

Hence we have, as in the proof of Taylor's theorem, 

F(a + h) _ F^'-'Xa + Ok) 
h:-' ~ \r-\ ' 

F(a + k) _ F^%a + eh) 
hr " \r_ ' 

which equations being always true, are true in the limit ; 

, F{a + h) _ F^-\a) 

- ,, F{a + h) F^^\a) 
and //^„--L__Z = __^ = OD. 

Now there will generally be some value of m such that 
^^A=o — j;ii, — shall be neither zero nor infinite; and in order that 

the above equations may be true, m must lie between r - 1 and r. 
We shall therefore have 

where C is independent of h and P a quantity which vanishes with 
h, and which, when arranged in powers of /«, must be of the form 

ra^+C'7i' + &c. 

.-. F{a + h) = C/r + C'h^^^ + CU'^^'^ + &c. 
or, putting for F{a + K) its value, 

f{a + h) =f{a) +fXa)h +/'-"(«) r^ + CA^ + C'/r^' + &c. 

i. e. the expansion follows the law of Taylor's theorem so long as that 
gives finite coefficients, after which a series of fractional powers of h 
commences, the first of which lies between the last integral index 
and the integer next greater than it. 



110 DEVELOPMENT OF FUNCTIONS. 

153. It will be observed that the term generally has been used 
in these demonstrations. The fact is, that our assumption that /(a:) 
may be expressed in the form (p(x){x — a)~"^j where (p{x) neither 
equals zero nor infinity when x = a, is not universally true ; and in 
these cases it is impossible to expand the function f(x) in ascending 
powers of the required quantity h. Thus, for example, if /(x) 
= log {x - a), f{a) = log = - CO, and lt^f{x) {x - ay = 0, for all 
values of m^ however small*; i. e. <p(a)=0 for all finite values of 
m, and = co when m = : there is therefore, in this particular case, 
no function such as we have supposed <^(a:) to be, which shall 
=/(x) (ar - a)"*, and yet be neither zero nor infinity when x = a. In 
this case the expansion of /(« + h), i. e. of log h in powers of kf 
cannot be found. 

154. Taylor's and Stirling's theorems enable us to expand all 
functions of one variable whose successive differential coeflBcients 
can be found. 

Expansions may sometimes be found more readily by the fol- 
lowing method. 

Let /(a;) be the function to be expanded in powers of a;. 

Assume /(a;) =A+Bx + Cx^ + 

then /W= B + 2Cj:+ 

/'(a:)= 2C+&C. 

Now if any of these differential coefficients are quantities which can 
be readily expanded by algebraical processes, or if any relation can 
be found between the quantities /(a:), f'{x)^ &c., it is evident that, 
by substituting for them the above values, we shall obtain an equa- 
tion for determining the indeterminate coefficients A, B, C, &c. by 
equating coefficients of equal powers of x. 

* This may be proved as follows: 

assume log(<r — a)= — y ; 

.'. y = co when ar = a 5 



7/t „ 



= -lL 



however small m may be. 



DEVELOPMENT OF FUNCTIONS. HI 

As an example of this method, let/(a;) = sin x. Assume 
sina? = u4(, + AiX + Azx'^ . » * + A^x"* + . . . 
or, as it may be written, 

sin X = 2^„af . 
Differentiating this equation twice, we obtain 

— sm X = 1.n {n - 1) An3c"~^j 
.-. = ^AX + 2w (w - 1) ^„a:"-^ 
The coefficient of ^""^ in the above is J.^2 + n(n'-l)A„, 
.*. A„_2+-n(n-l)A„ = 0, 

Also, since lt_^ = 1, Ao = 0, ^, = 1, 

and making n in the above equation successively 2, 3, 4 . . . 2m, 
2?w + 1, we obtain 

^0 + 2.1.^2 = 0, .-. Aa = 0, 

^x + 3.2.^3 = 0, .'. ^3 = -—, 

Asi + 4>.3.A, = 0, ,'. A^ = 0, 

A 2m_2 + 2m.2m - lA^^n = 0, .*. A^n, = 0, 

^2m-i + 2m + 1.2m ^2^+1 = 0, .*. ^2^+, =(-1)*" 



|2m + l 



^ x^ x' , ,,^ a;'"^' 



,\ sinx = x - — + —... +(-lY r~ -+.... 

[3 [_5 ^ ^ [2?« + 1 

155. To expand/(af + k, i/ + k) in powers of h and k. 
If we consider /(x, y) as a function of _y alone, and differentiate 
partially with respect to y^ we shall have, by Taylor's theorem, 
f{x,y + k) 

^, ^ df(x,y)y dy(x,y)P drf(x,y)k' 

•^ ^ *^^ c(y c?y' [2 d/ \r_ 

If in the above we write x + h for Xy we obtain 

f(a: + h,y + k)=f{x+k,y) 

df{x + k,y) d'fCx + h^yy k' jy{x-^-h,y) k\ 
dy "^ df L2---'" (// [rj'-' 

* The brackets are omitted from the partial differential coefficients for convenience, 
as no error can arise from so doing. 



112 DEVELOPMENT OP FUNCTIONS. 



Expanding, in like manner, each term of the above series, 

f^x+h, y) =f{x, y) + —j^^ h + ^^, -j^ - ^ dx- [^^••• 

df{ x^h,y) ^ dfjx.y) ^ dyjx, y) ^ ^dy{x,y) J^ ^ 

di/ dy dxdy '" dx!^^ dy \n — \ 



dy 
d'f{x + h 


<3) 


df 


drf{x + h, 


y) 


df 



<£feil)... + ^Z(:^_j^ 



dy^ dx^'-Hy^ \ 7i-2 



n-2 

+ ... 



dx^ 


dy' 


If-' 



Therefore, substituting these values in the above equation, we obtain 

l2 ( dx^ dxdy dy' J 

\n\ dx"" dx"-^dy \n-r rij"-'^Jy j 

+ 

It will be observed that the coefficients and the indices of h 
and k, in the quantities within the brackets, are those given by the 
binomial theorem. 

156. If in the above proof we had changed x into x + h before 
changing y into y + k, the coefficient of /i"~'^'■ would have been 
changed from 

1 d rf{x, y) ^^ _1_ dy{x, y) 
l^w-r dxT-'^df \n~r dy^dx""-' ' 

Hence djjx, y>i d^fjx, y) 

^^"^^ dx^-'df ~ dy^dx''-' ' 

or the order of partial differentiation is immaterial, a property which 
we have before proved. (Art. 1 37-) 

157. The limits of the remainder of the above series, after 
terms of the (ji- Vf^ degree in h and k^ will be given by putting 
X + Qh, y + Qk fov x and y in all the terms of the n^^ degree, and 
determining the greatest and least values of each term when 6 varies 
from to 1, 



DEVELOPMENT OF FUNCTIONS. 113 

This series, like that on which it depends, fails when any differ- 
ential coefficients become infinite for the particular value assigned 
to X. 

158. If we put h = dx, k = c/y, and u =/(a', y\ and consider 
X and y, independent variables, Taylor's series for two variables, 

gives, 

^/ , ,x J d^u d^u . . ,. 

f{x + /if y + k) = u + du +- — ... + -j — + ... m I?!/. 

as appears by Art. 1 43. 

159. With the same conventions as those employed in Art. 143, 
Taylor's theorem for one and for two variables may be written 

In like manner it may be proved that 

If the variables under the functional sign are made independent 
variables, all the above forms are included in the general expression 

f{x + dx, y + dy,,..) = e^/{x, y ...) 
d being the total differential. 

160. Lagrange's theorem. From the equations 

y = z-¥X(p{y) (1), 

"=/(j/) (2), 

to expand u in ascending powers of x. 
By Stirling's theorem, 

fdu\ /d"u\ x" p 



' ' dx I - x(p' (y) ^ ^^^ dz 
and therefore, if C7be any function of y, 



dx ^ ^'^ ^ dz 

H. D. C. 



= 0(y)^ (^> 



8 



1 1 -t DEVELOPMENT OP FUNCTIONS. 

Let U= u, therefore 5^ = * (^) ^ • 
Also, since £=/'(y)J. 

dV 
If therefore Fbe such a function of^ that -j- =<l>{y)f{y)7 

^y 

^, ^du dV 
d'u d dV d dV dj^,.dF). ,^. 

Now suppose _=^-^|0(^)-_|, 

rf-^__^r du) 

•*• da:"^'~dxd2"-'\'^^^'^ dzi' 
d" (dV:\ 






i. e. if the law of formation above assumed holds for -j— , it holds 

ax 

d^^^u . d'^u 

for -yn+i > ^^^> ^s it has been proved for -7-^, it holds generally. 



Hence we have «/„ =/(^), 



DEVELOPMENT OF FUNCTIONS. 115 

and the expansion becomes 

161. Laplace's theorem. From the equations 

y=^F{z + xcp{y)} (1), 

^=Ay) C2), 

to expand u in ascending powers of x. 

By Stirling's theorem, 

fdu\ fd"u\ J?" . . /. 

rrom(l), |=f'{2 + a;0W}{0W+x<|,'(3,)g}, 

•••rf^ = *W^ (^)' 

where C7 is any function of y. 

Let U=Uj and assume (p{i/)-j-=-j— where V is some function 

of J/; 

^ ^^ d (dF\^_d dV 
dx^ ~ dx \dz) ~ dz dx ' 



-iM'^yi]- 



By the same process as in Lagrange's theorem, we may shew 
from this equation and equation (3) that the successive differential 
coefficients are given by the same law, 

d''ti d"-' ( , .du\ 

••• 'd^^^-d^^v'^y^TzV 

Hence "o=/{^(^)}^ 






8—2 



116 DEVELOPMENT OF FUNCTIONS. 

162. Laplace's theorem may be deduced from Lagrange's as 
follows. 

Let z + a:<p(i/) = v, .-. y = F{y), 

u-=f{F{v)}, 
v = z-\-x(p {F(y)}. 

These equations are now in the form required by Lagrange's 
theorem, in which we have only to change /and <p into /i^ and (pF, 
when we obtain the expansion of u as given by Laplace's theorem. 



CHAPTER VIII. 

LIMITING VALUES OF INDETERMINATE FUNCTIONS. 
MAXIMA AND MINIMA VALUES OF FUNCTIONS. 



163. To determine the limiting value of a fraction which 
assumes the form -g. for a particular value of the variable. 

Let ^ , ; be a fraction whose numerator and denominator both 

vanish when ji; = a. 



By the definition of a differential coefficient, 



Ax 

. u f{x-^Ax)- f{x)_f\x) 

*' ^=' <p{x + Ax)-(p{xy <p'{x) ....\i;. 

In the above equation put x==a, 

\ . If /(« + A^) f'M 

If • ^ ,, { is of the form 5, this equation is indeterminate, but in that 
(p{a) ^ 

case we have, since equation (1) is true in the limit, 

■It l^-it /'w 

<p{x) <p(x) 

Let now the first pair of differential coefficients of the same 
order of /(j?) and (^(a*), whose ratio is determinate when a; = «, be 
/("^(a), (^*"^(«) ; then, by applying the above result to the differential 
coefficients in succession, we have 

^ (or) " ^^ (p'{x) ^^ (^'^"-^) (a:) ~ (^(") (a) • 

164. To determine the limiting value of a function which as- 
sumes the form § for a particular value of the variable. 



118 LIMITING VALUES OF INDETERMINATE FUNCTIONS. 



Let the fraction be ~7-i where /(o) = co, <^ (a) = 03. Then 



1 



dx { (p {x) ) 



. 7/ /W-7, /W" 

••"'"0w-""'^'(-^)' 

the same expression as that found when/(«) and (p {a) equal zero. 
If/' (a) and ^'(«) have a determinate ratio, this becomes 

,, /W../W. 

but if /'")(«), ^'''^(a) is the first pair of differential coefficients whose 
ratio is actual, we have, as before, 

165. If all the differential coefficients of/(.r) and ^(o:) have 
indeterminate ratios when x = aj this method fails to determine the 
limiting value of the fraction. In this case we can only obtain it by 
algebraical methods, as in the examples of Chapter II. We may 
proceed as follows ; 

If f^-Jf /(^ + ^) 

Let the expansions off(a+h), ^(a + h) in ascending powers of ^, 
obtained algebraically (since, when the differential coefficients become 
infinite, Taylor's theorem fails) be 

<p{a + h)=.A'hr' + B'}f'+... 
where m and m' must be negative or fractional. Then 

''=" <p (x) ~ '=' A'h-' 4- B'h"' + . . . ' 
_, Air-""' + Bh"-""' + . . , 
~ '='~A'VB'h'''-'^'^^:77 ' 
= if m is > m'j 

A .. 

= -j;- it w = m , 

= CO if ?« < m'. 



LIMITING VALUES OF INDETERMINATE FUNCTIONS. 119 

If we had employed this method when neither /(«), <p(a) nor any 
of the differential coefficients are infinite, we should have obtained 
the results of Art. 163, as is evident from the fact that the expan- 
sions will be those given by Taylor's theorem. 

166. There are some other indeterminate forms which a func- 
tion may assume, the limiting values of which may be made to de- 
pend on those of functions which assume the form ^. Such forms 
areO", co", 1=^®=. 

(1) Let w=/(a;)<l'W where /(a) = 0, 0(«) = O; so that when 
a; = a, u is of the form of 0". Then 

and when x=za, <p {x) log/Qjc) assumes the form O.oo or ^^ the limit 
of which can therefore be determined, and thence that of u, 

(2) Let f(a) = co, (p(a) = 0; then u assumes the form co° when 
x = a: but <p (a:) log/ (a;) assumes the form O.oo or .0., the limit of 
which, and thence that of v, can be determined. 

(3) Let /(«) = 1, (^a) = ± co. Then u assumes the form 1=^~ 
when x = a, and (x) log/ (a;) assumes the form co.O or q , the limit 
of which', and thence that of u, can be determined. 

Maxima and Minima, 

167. To find the maxima and minima values of functions of one 
variable. 

Dep. If, when x receives a continuous increase, /(j:) increases 
until X has a certain value, a, and afterwards decreases,/(a) is called 
a maximum value of f{x) ; and if /(^) decreases until x attains the 
value a and afterwards increases, /(«) is called a minimum value 
affix). 

Since /\x) equals the ratio of the rates of increase of /(a?) and x, 
it follows that when an increase of x produces an increase of f(x), 
/'{x) must be positive, and when an increase of x produces a de- 
crease of f(x')j f\x) must be negative. Hence when x, in the 
course of its increase, passes through a value, a, which makes /{x) 
a maximum or minimum, f\x) must change its sign from positive to 
negative in the former, and from negative to positive in the latter 
case. 



120 MAXIMA AND MINIMA VALUES OF FUNCTIONS. 

Hence at such a point f{a) must either equal zero or infinity. 

The converse of this however, viz. that all values of x which 
make f'{x) either zero or infinite correspond to maxima or minima 
values of /(a?), is not true; because a function, as /'(x), may pass 
through these values without changing its sign, which is the criterion 
bi f{x) having a maximum or minimum value. 

To determine, therefore, all the values of x which make f{x) a 
maximum or minimum, we must first find the values of x which 
satisfy either of the equations 

/'W = °' 7^=0 ••(»)• 

and for each of these values inquire whether f'{x) changes its sign 
in passing through it. If not, this value of x does not make f{x) 
either a maximum or a minimum : if there is a change of sign from 
positive to negative, f{x) has a maximum, and if from negative to 
positive, a minimum value. 

The existence and nature of the change of sign o^ f'(x) for any 
value, «, of X which satisfies either of equations (1) is easily deter- 
mined by observing the signs of f{a + h) and f{ a - h), when h is 
made so small that there shall be no change of sign in fQc) between 
x-a and x = a ■¥ h, or between x = a and x = a-h. 

168. The last proposition becomes self-evident when stated in a 
geometrical form. 

Let the figure represent a curve whose equation is ^=/(x) 
A 




T L M N 

having a maximum ordinate at A and a minimum one at B. Then 
if P be any point x, t/, tan PTL =f{x). From P to ^, the value of 
this tangent is evidently positive ; at A it passes through zero to a 
negative value, which it retains until at B it again passes through 
zero and becomes negative. The change from positive to negative 
therefore corresponds to the maximum, and that from negative to 
positive to the minimum value. 

The case where /'(a?) passes through infinity, will be considered 
geometrically in a subsequent chapter. (See Art. 209). 



MAXIMA AND MINIMA VALUES OF FUNCTIONS. 121 

169. The criterion by which we distinguish between maxima 
and minima is sometimes more convenient in the following form. 

If any value of a: makes /(a?) a maximum, /'(x) changes from 
positive to negative in passing through that value ; hence, in this 
case, an increase of a; makes f(^x) decrease at the point in question ; 
and similarly, if f(x) is a minimum, an increase of a; makes /'{io) 
increase. 

The differential coefficient o£ f\x) must therefore be negative in 
the former, and positive in the latter case ; i. e. if a be any value of 
X which satisfies either of equations (1), 

f{a) is a maximum if /''(a) is negative, 
minimum positive. 

This criterion fails when f'{a) is either zero or infinite : in the latter 
case we must proceed as in the previous article, in the former we 
may either employ that method or the following. 

Let the first differential coefficient of /(a:) which does not vanish 
when X = ay be /"(«). Then, by Taylor's theorem, 

\n 

and J^{a - h) -f{a) =^!^^^^ (- hy. 

But if /(«) is a maximum, both these quantities must be negative ; 
and if a minimum, both must be positive, if h is taken sufficiently 
small. 

Also, when h is sufficiently diminished, ho\h f^"\a + dli) and 
/'^"^a — dh) assume the same sign as /^"'(a). Unless therefore n is 
even, the above expressions will have different signs, and f(a) will 
be neither a maximum nor a minimum value; if n is even, the signs 
of both will be the same as that of /^"^(«),• 
.^ f(a) is a maximum if /*"'(«) is negative, 
minimum positive. 

Hence the rule. If the first differential coefficient which does not 
vanish is of an even order and negative, there is a maximum, if of 
an even order and positive, a minimum value; if of an odd ovdet 
there is neither a maximum nor a minimum. 

170. The above rule is equally true if we read differential for 



122 MAXIMA AND MINIMA VALUES OF FUNCTIONS. 

differential coefficient, and this, whatever may be the independent 
variable. For, from the expressions of Art 98, it appears that if 
y =f{^) and dy = 0, /(x) will also vanish ; and if all the differentials 
of y, up to d*~^i/ inclusive, vanish, the differential coefficients up to 
the (n — ly^ wiU also vanish. Again, if d"j/ is the first differential 
which does not vanish, for a particular value of x, as a, dy must 
(when a: = a) become equal to f^''\a)dx'', as well when the indepen- 
dent variable is general as when x is independent variable ; for by 
observing the equations of Art. 97, it is obvious that the first term 
in the expression for J^y will be /^"^(«)<7j:'*, and that all the other 
terms will be multiplied by lower differential coefficients which (as 
we have seen) vanish with the corresponding differentials. Hence, 
when n is even, the sign of d"i/ (for the value a of x) will be the 
same as that of f^"\a), since dx" must then be positive. The condi- 
tions dy = Oj d^7/= 0...(^~^{y) = 0, and c?"(y) a positive or negative 
even differential are therefore identical with those before obtained, 
viz. /{a) = 0, /'(a) = 0. . .f"^\a) = and /"\a) a positive or nega- 
tive even differential coefficient. 

171. To determine the maxima and minima values of w from 
the equations 

n=f{^x,y) (1), 

O^F{x,y) (2). 

This may be done by substituting in the former equation for y its 
value in terms of a?, when u becomes an explicit function of x and 
may be treated by our former methods. Those maxima and minima 
values corresponding to the condition du = (i may however be more 
conveniently obtained as follows. 

When w is a maximum or minimum, du = 0; 
.'.f{x)dx-^-f'{y)dy^O, 
also F' {x) dx + F' (y) dy = 0. 

By eliminating dy and dx between these equations, we obtain an 
equation which, together with (2), determines the values of x and y 
which make u a maximum or minimum. The sign of d^u will distin- 
guish them. 

If the same value of x which satisfies the above equations also 
makes d^u vanish, we must find the first differential, d'u, of u which 
does not vanish for that value of x, and observe whether n is even or 



MAXIMA AND MINIMA VALUES OF FUNCTIONS. 123 

odd, and in the former case whether d^'u is positive or negative. In 
doing this it will always be convenient, though not necessary, to 
make either x ox y independent variable. 

172. To find the maxima and minima values of a function of 
two independent variables. 

Def. If any values a, h, of the independent variables in the 
function f{x, i/) make f(a, b) always greater than f(a + h, h + k), 
whatever be the relative magnitudes and signs of k and k, provided 
they are taken sufficiently small, /(«, b) is called a maximum value 
of /(a:, 7/); and if/(«, b) is always less than /(« + ^, b + k), it is 
called a minimum value of /(jr, y). 

If/(«, b) is a maximum or minimum, the above conditions, which 
hold for all relative values of h and k, must hold for that particular 
system of values of h and k which satisfy the condition k = mh or 
y — b = m{x — d)', and conversely, if they hold for all such systems 
of values, when m is varied in all possible ways, they must hold for 
the general values, since there is no pair of values of A and k which 
cannot be found in some one of the particular systems. But the 
supposition y—b =m (x — a) reduces f(x, y) to a function of one 
variable, and our definition to that of a maximum or minimum value 
of such a functiori. 

Let therefore ^ be a function of one independent variable deter- 
mined by the equations, 

^=/('^,i/) (1). 

y — b = m{x — a^) (2). 

The conditions that z shall be a maximum or minimum are (by 
the previous articles). 

d2=f{x)dx+f(y)dy=^0 (3), 

and dy — m dx, 

whence dz = dx{f{x) + mf(y)} = (4). 

In order that f(x, y) may have a maximum or minimum value, 
this equation must hold for all values of m, which cannot happen 
unless both the equations, 

/W = o,/'(^) = o, 

are satisfied. Hence if /(«, b) is a maximum or minimum value, 
a and b must be a pair of roots of these equations. 



124 MAXIMA AND MINIMA VALUES OF FUNCTIONS. 

2 will be a maximum or minimum according as the sign of c?*2 is 
negative or positive for the above values of a; and 7/. If we differen- 
tiate equations (1) and (2) a second time with a: as independent 
variable*, we obtain from equation (2) d^i/ = and then from 
equation (1), 

d^z=f"{x)dx^ + of"{x,y)dxdy^f{y)dy (5) 

= dx^{r{x)+^f'{x,y)m^f'\y)m^} (6). 

When x = a, y^^t the value of d^z becomes, 

d<^ {f"[a) + 9.f"{a, h) m +f'(b) m'}, 

l£ /(a, h) is a maximum the above quantity must be negative, 

and if a minimum it must be positive, for all assignable values of m. 

In both cases therefore, it must be incapable of changing its sign for 

any change in the value of 7w, which will be the case if the equation 

f"{a)-^-'^f"{a,h)m+f"{h)m^=^0 (7), 

has impossible roots, that is, if 

f"{a)f"{b) is > [f"{a, b)Y. 
This is called Lagrange's condition. 

If it is satisfied, /(«, h) will be either a maximum or minimum: 
to discriminate between them, we observe that the sign of the above 
expression, being always the same, must be the same as that which 
it has when m = that is the same as that of f"(a), which again 
must be the same as that of f"{h), since otherwise Lagrange's con- 
dition would be impossible. Hence /(«, 6) will be a maximum or 
minimum, according as f"{a) and f"{h) are negative or positive. 

If Lagrange's condition is not satisfied the equation (7) will have 
real roots and the value of d'Z, when x = a and y = h, will in general 
be negative for some values of m and positive for others. 

Hence, some values of m will make z a maximum and others a 
minimum, in which case f(a, h) will be neither a maximum nor a 
minimum value of f(x^y). 

To determine therefore, the maxima and minima values of f{x,y), 
we must find all the roots of the equations 
/'W = 0, and/'(3.) = 0, 

• jr is made independent variable for simplicity's sake, but we might have left the 
independent variable general, when two additional terms would have been added to 
(5) which would have vanished when a, h were substituted for x and y, leaving the 
subsequent steps the same as in the text. 



MAXIMA AND MINIMA VALUES OF FUNCTIONS, 125 

which satisfy Lagrange's condition and then distinguish the maxima 
from the minima values by observingthe sign either o?f"{d) or of/"(6). 

173. There is one case in which f{x, y) may have a maximum 
or minimum value even though Lagrange's condition fails, viz. where 
equation (7) has equal roots, and therefore 

f"W\h) = {f{fl,h)\\ 

For if the roots be equal to m^ , d^z will retain a constant sign for 
all values of tw, except m^y and for that value d^z will equal zero. 

For the value Wj, of wz the condition that z may be a maximum 
or minimum will be that the first differential, dH'z, which does not 
vanish shall be of an even order, and then z will be a maximum or 
a minimum according as the sign of dl'z is negative or positive. 

If therefore, n is even and the sign of c?"^ when m — m^^ the same 
as that of d^% for all other values of m, z will be a maximum for all 
values of m including m^ or a minimum for all such values. In this 
case therefore /(a, h) will be a maximum or minimum accordingly. 

But if n is odd, z will be neither maximum nor minimum when 
m — m^f and if w is even and the sign of d^'z different from the sign 
of d^z for other values of m, z will have a maximum or minimum 
Value when m — nii corresponding to minima or maxima values re- 
spectively for all bther values of m. In neither of the latter cases 
therefore will/(«, h)he a maximum or minimum. 

Again, if d'z vanishes for all values of w, which will be the case 
j£ /'{a) = 0,f'{a,b)=0, and/'(6) = 0, the condition that /(a, 6) 
shall be a maximum or minimum value, will be that the first actual 
differential d"z shall be of an even order and retain a constant sign 
for all values of m. The condition which will then take the place 
of Lagrange's will be that the equation of n dimensions in m, 

d'z = 0, 
shall have no real roots, with further conditions in the case of pairs 
of equal roots analogous to those above found. Such conditions are 
too complicated to be of much practical value, and in the majority 
of cases, Lagrange's condition will be found to apply. 

174. The hypothesis introduced in Art. 172, that m is an arbi- 
trary constant in the expressions (4) and (6), is evidently the same as 
the assumption that dx and dy are arbitrary constants in (3) and (5), 
in which case those expressions become the first and second total 



126 MAXIMA AND MINIMA VALUES OP FUNCTIONS. 

differentials of /(ar^ y) considered as a function of the two independ- 
ent variables x and y. The higher differentials of z are in like 
manner converted by the same hypotheses into the total differentials 
o? f{x,y) with X and y as independent variables. The conditions 
for determining the maxima and minima values of f(^x, y) may 
therefore be stated as follows : 

(1 ) df(x,y) = 0, which invol ves /' (x) = 0, f (t/) = 0. 

(2) d^f{x^ y) must retain a constant sign for all values of the 
differentials, the negative sign corresponding to maxima and the 
positive to minima values. 

(3) If d^ f{xy y) vanishes for the same values of x and y as 
df{xy)i the first actual differential must be of an even order, and 
must itself retain a constant sign for all values of dx and dy ; and 
if d^f{x, y) vanishes only for particular relative values of dx and dy, 
retaining a constant sign for all others, the first actual differential 
for those particular values must be of an even order, and must have 
the same sign which d^f{xj y) has for other values of dx and dy. 

(4) In every case a negative sign corresponds to a maximum 
and a positive to a minimum value. 

By a process similar to that of Art. 172, these conditions may be 
extended to functions of any number of independent variables. 

175. The geometrical significance of Art. 172 appears thus. 

Let a surface be constructed whose equation is z=f{x,y)^ and 
let/(«, h) be a maximum or minimum ordinate of the surface. Let 
a plane be drawn through this ordinate, making an angle Xjaxr^m 
with the plane of xz. Its equation will be equation (2) of Art. 172, 
and it will intersect the surface in a curve determined by the pair of 
equations (l) and (2). By making the intersecting plane revolve 
round the ordinate f{a, 6), i. e. by giving to m all possible values, 
we obtain a system of curve sections in some one or other of which 
every point of the surface is included. An ordinate which is greater 
than any of the surrounding ordinates of the surface, must be greater 
than the neighbouring ordinates in any of the curve sections, and 
conversely if it exceeds the neighbouring ordinates in all the sec- 
tions, it must exceed all the surrounding ordinates of the surface ; 
that is, if /(«^ b) is a maximum in all the sections it will be a maxi- 
mum also in the surface, and similarly, if it is a minimum in all the 
sections it will be one also in the surface ; but if the ordinate is a 



MAXIMA AND MINIMA VALUES OF FUNCTIONS. 127 

maximum in some sections and a minimum in others, it will be 
neither a maximum nor a minimum ordinate of the surface, since 
some of the surrounding ordinates will be greater and others less 
than it. Lagrange's condition is the analytical expression of these 
facts. When it is satisfied the ordinate will be a maximum or mini- 
mum in the surface, if it is so in any one of the sections. The most 
convenient sections to examine are those parallel to the co-ordinate 
planes, and this is what we have done, in employing the sign off" (a) 
or/"(6) as the test for the purpose. 

176. In the preceding articles we have considered only those 
maxima and minima values of functions of several variables, which 
make the differentials vanish. There are others which make them 
infinite, but they are of comparatively little interest except with 
reference to the singular points of surfaces, which is a subject quite 
beyond the scope of this work. 

177. To determine the maxima and minima values of a function 
of n variables connected by r equations. 

Let u = F(xi, .Ta . . . ar„) = max . or min., 

■where the variables are connected by the equations 



(1). 



Then, since u is a maximum or minimum, 

du = F\xi) dx, + F'{x^) ^2 . . . + F\x^) dx^ = 
also df, =//{x,) dx, +f/(x,) dx,... +// W dx, = o] 

df, =// W dx, +//W dx, ... +/;(^„) dx,= 0^ 

From these (^ + 1) equations we may eliminate r of the differ- 
entials, leaving a linear equation involving the remaining {n — r) 
differentials. This elimination may be conveniently effected by the 
method of indeterminate multipliers. 

From the systeni of equations (1), we obtain 
du + Xif^i + Ajc^g . . . + \df, = 0, 
which holds for any values of Aj, Ag ... A^. 



128 MAXIMA AND MINIMA VALUES OF FUNCTIONS. 

If we give to du, dfx ... df, their values from equations (l) and 
collect the coefficients of each differential, this equation becomes 

3f 1 dxx + M2dx2 . . . + M^doc,, = 0, 
where each quantity M is of the form 

F'{^) + \A'{x) + X,f/(x) ...+ KfJ{x). 

The multipliers Ai, Ag ••• ^rj being indeterminate, we will give them 
such values as to satisfy the r equations 

JVfi = 0, M^ = ...Mr = (2), 

by which assumption the above equation is reduced to the form 
M^i dx^x + Mr+2 dxr+2 •..+M„dx„ = 0. 

Now in the given system of equations there are (n — r) indepen- 
dent variables, and therefore the (w — r) differentials in this equation 
are perfectly arbitrary, and the equation cannot be satisfied unless 

M,^i = 0, M,+s = . .. M„ = 0. 

By substituting in these equations the values of A,, Ag ... A^, obtained 
from (2), we have (n — r) equations, which, together with the (r) 
given relations among the variables, are sufficient to determine the 
values of a?!, x^ ... x„, which make u a maximum or minimum. 

Hence, whatever be the number of dependent and independent 
variables, we have the (n + r) equations 

fx(xi, X2 ... x^) = 0, 
/2(^i, x,...x„) = 0, 



f{xi, X2 ... x„) = 0, 
F'ix,) + A,// (or,) + A,/,'<x,) ... + \fr'{x,) = 0, 
F'ix^) + A,//(a:,) + A,//(x,) ... + KfJ{x,) = 0, 

F'{x:) + A,//W + A,//(^„) ... + xj/(x„) = 0, 

involving the (ra + r) quantities x^, X2 ... .r„, Aj, Ag ... A,, from which 
we may determine the values o£ x^ Xz ... x^, which make u a maxi- 
mum or minimum. 

The introduction of interminate multipliers, makes this investi- 
gation rather complex. The principle involved is however very 
simple. It is (as may be seen) merely this. First to apply the test 
of a maximum or minimum value, viz., du = 0, and then after elimi- 



MAXIMA AND MINIMA VALUES OF FUNCTIONS. 129 

nating as many differentials as possible to make the coefficients of 
the remaining independent differentials separately vanish. In most 
examples the use of indeterminate multipliers will be found much 
simpler than any other mode of elimination. 

The investigation of the condition which in this case supplies the 
place of Lagrange's condition, and the application of the test to dis- 
tinguish maxima from minima values, would be extremely trouble- 
some. In most problems, however, it is easy to see from a priori 
considerations whether maxima or minima values exist, and to dis- 
criminate them from one another, the exact values of the variables 
only remaining to be determined. In such cases the investigations 
alluded to become superfluous. 

178. Ex. 1. Find the maxima and minima values of a;^(a-j:)'. 
u = A'^ (a - xf, 

-j- = x^{a- x) [Sa - 5x] = or oo . 

The only values of a: which satisfy this condition are 
x = 0; or = a, and x = ^a, 

except a: = OD , which renders u also infinite, and need not be consi- 
dered. 



When X passes through the value 0, 
du 
di 



■J- changes from h^ {a + h) {Sa + 5h) 



to k\a-h) (3a -5k), 
both of which have the same sign when k is sufficiently small. 
Hence when x—0, wis neither a maximum nor a minimum. 

When a; passes through the value a, 

-T- changes from - (a- hyk(2a — 5K) 
ax 

to {a-\-hyh(2a + 5K), 

the former of which is negative and the latter positive, when h 
is sufficiently small. Hence when x= a, u has a minimum value 
equal to zero. 

When X passes through the value J a, 

-7- changes from {\a - Kf (^a + h) h 
to - {^a+hy (la -h)h, 

H. D. C. 9 



130 



MAXIMA AND MINIMA VALUES OF FUNCTIONS. 



the former of which is positive and the latter negative when h is 
sufficiently small. Hence, when x = fcr, u has a maximum value 

equal to~{\a)\laf or -^a\ 

The same results may be obtained by the method of Art. 1 69. 
Since 

■j- = x^{a — x) (3a — 5x)j 
ax 



dx 



^ = ^^(a-:r)(S«-5x){?-^-g^}, 



j-^ = 2{a -x){3a-5x)+P, 

where P is a quantity which vanishes with x ; 

Therefore when a; = 0, -^-g = 0, -7-3 = 6a^; 

that isj the first differential coefficient which does not vanish is of an 
odd order, and therefore u is neither a maximum nor a minimuin. 

d^u 

When a: = a, -r-^ = 2a^, which is positive, therefore w has a mini- 
mum value. 

When X = fa, -r-2 = - 2 (f )^a^ which is negative, therefore w has 
a maximum value. 

Ex. 2. To find the maxima and minima ordinates of the lem- 
niscate. P' -p 

The equation is 

(x' + fy = a'{x'-f) (1), \^^ 

and its form is that of the figure. 
By differentiating (1) we obtain, 

2 (x^ + j/^) {xdx +y dy) = a^ {x dx-y dy), 




M 



whence x^ = -^ — ^r~^ — ^-• 
dx a + 2x^ -f 9.y y 

The equation -^ = will be satisfied when 



9 9 



(2)- 



MAXIMA AND MINIMA VALUES OF FUNCTIONS. ISl 

The values of x and y obtained from this equation together with 
(1) are 



wl"' 



•^ = *'V 8 

which may therefore correspond to maxima or minima values. To 
distinguish them we will apply the criterion of Art. I69. 

By differentiating -^ , we obtain 

when P includes only terms which vanish when equation (2) is 
satisfied. 



Hence when « == ± a/ - a, and y = -\- ^ - a, 



d^y _ 3 
dx'~ J2a' 

which corresponds to a maximum value. 

If therefore Sm = OM' = V? «, and MP = M'P' = V^ a, 

MP and M'P' are maxima ordinates, 

when x=^j^~a, andy=-^^ -a, 

d'y 3 
^•^^ ~ J 2a ' 
or y has minima values at the points Q and Q,\ 

Since a maximum value of (- y) corresponds to a minimum value 
of y, the geometrical ordinates MQ, M'Q! are maxima. 

The values a? = ± a and y = will make -p infinite, but they do 

not correspond to maxima or minima values, since x cannot exceed 
a without rendering the equation (1) impossible. 

In the following example we shall employ the method of indeter- 
minate coefficients. 

9—2 



132 MAXIMA AND MINIMA VALUES OP FUNCTIONS. 

Ex. 3. To find the rectangular parallelopiped which shall con- 
tain a given volume under the least surface. 

Let the edges be x, y, z, d the given volume. Then 
u — xy-^xz-\-yz = min. 
xyz = a^, 
.*. {y -^ z)dx-\-{x^ %) dy-v{x-¥ y) dz = 0, 
yz dx + xzdy + xy dz = 0, 
.'. (y + z + Xyz) dx + (x + z + \xz) dy + {x + y + \xy) dz = 0. 

.*. y + z + Xyz = (1), 

x + z + Xxz = (2), 

x+y + Xxy = (3). 

(1) a;- (2)^ gives z{x^y) = 0, 

similarly x(y — z) = 0, 

and y(z- x) = 0, 

These, together with xyz = a?^ are satisfied hyx = y-z-a\ that 
is, the parallelopiped must be a cube. 

It is obvious that this form makes the surface a minimum^ since, 
by diminishing one side, we may increase the surface as much as we 
please. 

Examples will be found in Gregory's ExampleSj Chap. vii. 




CHAPTER IX, 

TANGENTS AND ASYMPTOTES. 

179*. Def. Let APQ be any curve, TPQ a straight line 
cutting it in the points P and Q. y 
Suppose Q to move along the curve 
towards P, then the direction of 
T'PQ will continually change, and 
it will approach a certain limiting 
position as Q approaches P. Let 
TP be this limiting position ; then 
TP is called the tangent to the curve APQi at the point P. 

A line PG drawn through P perpendicular to the tangent at P, 
is called the normal to the curve at the point P. 

180. To determine the inclination of the tangent to the co- 
ordinate axes. 

Let the equation to the curve referred to rectangular axes Ox 
Oy, be y =/C^), 

OM = X,] MP = y, co-ordinates of P, 

ON =x + Ajjj NQ=y + Ay, co-ordinates of Q. 
Draw PR parallel to the axis of j:; then 

PR = Ax, RQr^Ay, 

.-. tanPr. = ^ = ^; 
PR Ax' 

and since PT is the limiting position of QPT' when Q approaches 
P, that is, when Ax and Ay approach zero, 

tan PTx= It^^ tan PTx = /^a=o tt » 



Ax 



.'. tanPra: = ^, 
ax 



-^ is known from the equation to the curve, and the inclination of 
ax 

the tangent to the axis of x is determined. 

* The substance of this and the following article has already been given by way 
of illustration. It is repeated here in a more condensed shape, as it properly belongs 
to the subject of this chapter. 



134 TANGENTS AND ASYMPTOTES. 

Hence, if x\ y* are the current co-ordinates of the tangentj its 
equation is 

y'—y_x'—x 

dy dx 

Since the normal is perpendicular to the tangent, 

dx 
tan PGfa: = - cot Pr^ = - J- ; 

dy 

therefore the equation to the normal is 

y-y—Ty^"^-^^^ 

x' — X v'—y 

or — i — + -^ -^ =0. 

ay ax 

In the figure of Art. 179, ^T, MG are called the suhlangerd 
and subnormaL 

Now Mr=MPcotPr^, MG = i)fPtanPra:, 

.*. subtangent = y -T- , 

subnormal = y-f-. 
^ dx 

181. The sine and cosine of the angle PTx may be conveniently 
expressed as follows. 

Let the arc ^P = *, arcPQ = Aj, chord PQ = c; then 

PQ 
PQ 



sin PTx = //a=o sin PTx = It^ 



-lU ^^ 



Now //a=o -^^J- ^^^ ^^A=o — = 1. by Art. 1 1, 

.-. sinPTx^^; 
ds 

PR 
also, cos PTx = /^A=o cos PT'x = //a=o p^ 

,^ A^ A* 
= ^^^-A7T' 

.-. cos pro: = ^; 
a* 



TANGENTS AND ASYMPTOTES. 135 

hence (^Y + (^Y = cos^ PTx + sin^ PTx = 1, 

or dx^ + dy^ = ds^. 

182. The lengths of the portions of the tangent and normal 
between the point of contact and the axis of x, are given by the 
equations 

FT = MP co^ec PTx 

_ds^ J{dx^+df) _ J{l+f(xy} 
-y dy y dy -y fix) ' 

PG=MPsecPTx 

The expressions for MT^ MG,PT,PG, may also be immediately 
obtained from the equations to the tangent and normal. 

Thusj putting y = in the equations to the tangent and normal, 

we obtain in the two cases 

, dx 
x-x =y-rr , 
^ dy 

, dy 

Hence, in order that our expressions for the subtangent and 
subnormal may hold in all cases, we must reckon MT positive when 
T lies on the negative side of M, and MG positive when G lies on 
the positive side of M. In the figure of Art. 179 these quantities 
are positive. 

183. To determine the concavity or convexity of a curve at any 
point. 

A curve is convex to the axis of x, so long as the inclination of 
the tangent to the axis of x increases with x, and concave when it 
decreases with an increase of x. 

In the former case, therefore, the differential coefficient of 
tan PTx must be positive, and in the latter negative. 

Now, tan PTx= -— when^ is positive. 



dx 



—■ when y is negative ; 



136 TANGENTS AND ASYMPTOTES. 

therefore the curve is convex or concave to the axis of jc according as 
-T^ and 1/ have the same or different signs. 

184. To determine the position of points of inflexion. 

Def. a point of inflexion is a point where the direction of the 
curvature changes from convex to concave, or from concave to 
convex. 

At such a point the inclination of the tangent to the axis of x, after 
increasing up to the point of inflexion, begins to decrease, or vice 
versa. The inclination therefore, and consequently its tangent or 

-^ , must be a maximum or a minimum at a point of inflexion, the 

analytical condition of which is 

-^=0 or CO. 
dxr 

If not only -~^ ^^^ also -7^ vanishes, we must have the addi- 
tional condition, that the first differential coefficient which does not 
vanish shall be of an odd order. 

The values of x which make -j^ infinite, will not necessarily cor- 
respond to points of inflexion, since this condition may be satisfied 
without the inclination having a maximum value. 

When the tangent is parallel to the axis of ^, -^ , and therefore 

d^u . 

generally -7^ , are infinite, whether the point is a point of inflexion 
ux 

or not. In this case we may employ the condition ^-^ = or co , in- 
stead of the above, as it might have been obtained in a precisely 
similar manner, and is free from the ambiguity attaching to our 
former criterion. 

185. To find the equation to the asymptote of any curve. 

Def. The asymptote of a given curve is a straight line or curve 
which the given curve continually approaches but never meets. If 
the equation to the given curve is algebraical, the asymptotes may in 
general be determined as follows. 



TANGENTS AND ASYMPTOTES lot 

Let the equation to a curve be f{j>c,y) = 0; and let this be 
reduced to the form 

y = Ax''+Bx^,.,+ G+Hx-''+SiC (l), 

the indices being arranged in descending order. Let the equation 
to another curve referred to the same axes be 

y = Ja?" + ^a;'»...+ (? (2), 

then {y - y') = Hx-" + &c., 

i. e. the distance between the two curves is generally finite when x 

is so, but approaches zero when x approaches infinity. 

The curve (2) is therefore the asymptote to the curve (I), 

If w = 1, w = 0, the equation (2) becomes 
y' = Ax-¥ B, 
and the asymptote is rectilinear. 

If in the equation to the curve, y is given explicitly in terms of 
X, the expanded form may be obtained by the ordinary algebraical 
methods. If y is only given as an implicit function of x, the ex- 
pansion must be found by means of indeterminate coefficients. 

The most important cases are those in which the asymptotes are 
rectilinear. 

It is evident that a rectilinear asymptote coincides with the limit- 
ing position of the tangent when one or both of the co-ordinates 
approach infinity. 

This is sometimes given as the definition of an asymptote, and 
in many cases the asymptotes may be readily determined from this 
consideration. 

Whenever there is a rectilinear asymptote, the perpendicular upon 
it from the origin, and therefore one at least of its intercepts on the 
co-ordinate axes, must be finite. Hence the test of the existence of 
an asymptote is, that the intercepts of the tangent shall not both 
become infinite when the co-ordinates of the point of contact are put 
equal to infinity. 

Asymptotes parallel to either of the co-ordinate axes may be at 
once discovered, by observing whether any finite value of one of the 
co-ordinates renders the other infinite*. 

186. Examples on the preceding articles. 

(1) Let the equation to the curve be 
x^ v^ 

-K^-%-^ (')'• 

* Symmetrical investigations of points of inflexion and asymptotes will be found 
in the Mathematical Journal for February 1841 and November 1843. 



138 TANGENTS AND ASYMPTOTES. 

xdx _y dy dy _h^ x 

and the equation to the tangent becomes 

x{x'-'x) _ y{y'-y) 

^'-^ = lby(l). 
The equation to the normal is 



xx' yu 
or — --^'^ 

a 



or a^yx' + V^xy — {a? + ^^) xy. 

The equation to the asymptote may be found from that to the 
tangent by making x and y approach infinity. The equation to the 
tangent may be put into the form 

V — —9 — X . 

Now when x and y approach infinity — approaches zero, and by (1), 



r'VC-j?- 



which approaches ± j when y approaches infinity. Therefore the 
equation to the asymptotes is 

This may also be found by expanding y in descending powers of 
X. Thus 

3^ = =^^n/(^^-«^). 



a 

= =t 



M-*^--}. 



b . ab 
a ^ X 



Hence, by our definition, the equation to the asymptotes is 



Since y~y'='^i — + 

X 



TANGENTS AND ASYMPTOTES. 1S9 

the sign of (j/-y'), when x is positive and sufficiently large, is nega- 
tive for the asymptote whose equation is y'= +~ x', and positive for 

that whose equation is y = x' ; that is, in the former case the 

curve lies below the asymptote when x is positive, and in the latter 
above it. Since the sign o^ y—y' changes with Xy the reverse will 
be the case when x is negative. 

This is easily seen to agree with the known form of the hyperbola. 

(2) To find the asymptotes of the curve 

{f-vhf){x-a)-x'^0 (1). 

When X approaches a, the value of y^ + hy"^, and therefore that of 
y, approaches infinity, and there is therefore an asymptote parallel to 
the axis of y at a distance a from it. 

To find any other asymptotes that may exist, we must expand y 
in a series of descending powers of a;. Assume, therefore 

Q 

y = Ax + B+—-{- &c., 

X 

.-. / = ^ V + SA'Bx'' + &c., 
.-. {x - a)f = ^ V + {^A^B - a A') x^ + &c., 
b{x-a)y^= hAW+Scc, 

-x'^-x', ' 
Adding these quantities, we have, by (1), 

= x*iA''-'i) +x^ {SA'B - aA^ +hA') + SiC (2), 

which must hold for all values o? x; 

.-.^'-1 = 0, .-. J. = l, 

3A'B-aA^+bA' = 0, .-. B = '^^. 

o 

Therefore the expanded form of the equation to the curve is 

a-h G ^ 
i/ = a: + -^+- + &c.,.. 

and consequently that to the asymptote, 

a-h 



y' = x' + 



S 



If it had been required to determine whether the curve lies 
above or below the asymptote, we must have collected the coefficient 



140 



TANGENTS AND ASYMPTOTES, 



of one more term in the expansion (2), by equating which to zero 
the value of C might have been determined. 

The relative position, however, of the curve and asymptote in 
such cases may generally be more easily determined by indirect con- 
siderations, as will be seen when we consider the methods of tracing 
curves. 

(3) To find the points of inflexion of the curve y = ^^ — j~- . 

(x-aj 

Then dy _ S{a:-aY 

which vanishes when a? = a ; and as the next differential coefficient 
does not vanish when x = a, this point is a point of inflexion. Since 

y and -^ also vanish when x = a, the tangent at the point of inflexion 

is the axis of or, and the form of the curve is that of the figure 



Curves referred to polar co-ordinates. 

187. To determine the angle between the tangent and the 
radius vector of any curve. 

Let QPT' be a secant through two 
points P, Q of a curve APQ. FT the 
tangent at P ; r, 6 co-ordinates of P, 
r + Ar, 6 + Ad co-ordinates of Q, refer- 
red to a pole S, and prime radius Sx, 
Draw PR at right angles to SQ. 

Then, by the definition of a tangent, 
tanSPT= ltpQ=o tan SQT', 
PR 




It 



PQ=0 



RQ' 



TANGENTS AND ASYMPTOTES, 141 

rsinA0 
"^=V + Ar-rcosAt^' 

rAg sin A6 1 

^=^ A^ "^^a~, ^ sin^iAf?* 



And ^^^=o^=r^, 

7/ !!!L^-i 
^="~Aa~" ' 

. Ssin'^lAa ,^ siniA0Aa .' . . ^ 

.-. UnSPT=:rf, 
dr 

If we call APj s, and therefore PQ, A^, and the chord PQ, c, 
we have 



Arf l + Sr 



Ar / ^s 



since /i!^_o — = 1. 

c 

P7? 
Also sin SP T ?= //pQ=o -- , 

c 



As c 

_dr 
"ds' 
As 



__ J rAd sin Ad As 

de 

ds " 



or dr' + r'dd' = ds\ 

188. If ST, SY are drawn at right angles to SP, Pr respec- 
tively, we have 

ST = r tan SPT, 






«Sr is called the polar sub tangent. 



142 TANGENTS AND ASYMPTOTES,. 

Also, if SY=p, 

p = r sin SPT, 

This quantity may be conveniently expressed in terms of the 
reciprocal of r. If this be w, we have 

, du 

^^ = -^- 

^, - du^ d6' 

i- 4^ 



-e 



It may also be expressed in terms of the rectangular co-ordinates 
as follows. 

If S is the origin and Sx the axis of .r, we have 

a = tan-^^; 
x' 

x^ + f 
xdy — ydx 

r^dd xdy —yd£ 
' ' ^ ds ds 

The expression for ST has been found from a figure in which 

dr 
r, 0, -7^ have all positive signs, but it applies equally to all cases, if 

we interpret properly the sign with which it is aifected. By con- 
structing the figures in the cases where any of these quantities are 
negative, it is easily seen that the direction of ST is always deter- 
mined by the following rule, when the positive direction of d is 
taken to the left of the prime radius. Loolc alofig the positive direct 
tion of the radius vector, and draw ST to the right or left, according 
as it is positive or negative. 

It may also be observed that the expression for tan SPT is, in 
all cases, positive when the tangent falls behind the radius vector, 
and negative when on the other side of it. 



TANGENTS AND ASYMPTOTES. 143 

189. To determine the asymptotes of polar curves. 

When any value of 6 renders r infinite, there may be a recti- 
linear asymptote. This will be the case if the polar subtangent re- 
mains finite when r becomes infinite. Since in this case the angle 
SPT vanishes, the asymptote will be parallel to the infinite radius, 
and at a perpendicular distance from it equal to the corresponding 
value of STj and measured in the direction indicated by its sign. 

When the radius vector approaches a constant limiting value, as 
6 approaches infinity, the curve will approximate to a circle which is 
called an exterior asymptotic circle when r increases in magnitude up 
to the limiting value, and an interior asymptotic circle when r de- 
creases towards it, as in the former case the circle is without, and in 
the latter within the curve. 

190. To determine the concavity or convexity of curves referred 
to polar co-ordinates and the position of points of inflexion. 

It is easily seen that when a curve is concave to the pole, p 
increases with r, and when convex p decreases with an increase of r. 

The curve is therefore convex or concave according as -^ is nega- 
tive or positive. 

At a point of inflexion, the curve changes from convex to concave 

or vice versa, and therefore -^ changes its sign. At such a point 

therefore we must have 

dp ^ 

-f- = 0, or CO . 

dr 

Those values of r which satisfy this condition, and also produce a 
change in the sign of J- , will correspond to points of inflexion. 

Examples will be found in Gregory's Examples, Chap. ix. 



CHAPTER X. 

CONTACT OF CURVES. 

191. Let y=f{pc), q/ =^ F{x) be the equations to two curves; 
then if, when a certain value, o, is given to x^ the values of the 
ordinates of the two curves are the same ; that is, if 

the two curves have a common point whose co-ordinates are a, /(«)? 
and are said to intersect. If at the point of intersection we have also 

f{.a) = F'{a), 
the curves have a common tangent, and are then said to have a con- 
tact of the first order. 

And generally, if we have 

f{a) = F{a\ f{d) = F\a), f{a) = F"{a),..f\a) = F^Xa), 
the curves are said to have a contact of the -nP^ order at the point in 
question. 

The reason of this definition appears from the following pro- 
position. 

192. If any curve has a contact of the wP^ order with a second 
curve, and a contact of the w*^ order with a third at the same point, 
where m is greater than n, and an ordinate is drawn cutting the 
three curves, the portion intercepted between the curves whose con- 
tact is of the wP^ order bears to that intercepted between those whose 
contact is of the w*^ order, a ratio which approaches zero as the ordi- 
nate approaches that through the point of contact ; or, as it may be 
expressed, the curves whose contact is of the higher order are, in the 
limit, infinitely closer than those whose contact is of the lower order. 

For, let the curve y =f(jx) have a contact of the m^^ order, and 
yT=(p{x) Si contact of the w*^ order, with y = F{x) at a point whose 
abscissa is a ; let an ordinate be drawn to the three curves whose 
abscissa is a + A, and let A^, A„ be the portions intercepted between 
y = F(x) and the two other curves. 

Then ^^=f{a + h)-F(a + h\ 

and /(a + h) =f{a) +f{a)h ... +/-'(«) ^ +f--\a -f dk) ^'"' 



Im'^ -■--[ 



m 



CONTACT OF CURVES. 145 



F(a+h) = F (a) +F'{a)h...+ F^-\a) — + F^"^'Xa + Oh) 



[rn ^ ^ [tw + 1 ' - 

The first (m+ 1) terras of these series are equal by the conditions 
of contact of the 7n^^ order, 

Lm+\ 



^^^^^ {f--\a + eh) - F^-^^^(a + eh)} ; 



n+l 



similarly, A„ = -^-^ {^("+')(a + eh) - F^^\a + eh)}, 

• ^-- h--^ ^" '^ ^ - ^"''^'^'' ^^^ " ^'"^'^^ "• ^^^ 
• • A„ [ m+1 (^("+X« + ^^0 - ^'"^'*(« + ^^) * ' 

When h approaches zero, the coefficient of h"^'* approaches a finite 

value, since otherwise the contacts would be of higher orders than 

those supposed ; and if m is greater than w, the limit of /i*"~" is 

zero, 

193. Curves which have a contact of an even order cut, and 
those which have a contact of an odd order touch without cutting at 
the point of contact. 

For if A^ is the excess of the ordinate o£ 7/=/(x) over that of 
y = F{x) with which it has a contact of the m^^ order, 

^ {/->(« + eh) - J'l-Xa + 0h)}. 

When h is made sufficiently small, the sign of the coefficient is 
that of/''"+^*(a) — F''^'^^\a) whether h is positive or negative, and the 
sign of h"^^ changes with that of A, when m is even and is unaltered 
by a change of sign in h when m is odd. In the former case 
therefore the ordinate of one curve is in excess on one side of the 
point of contact, and that of the other on the other side, while 
in the latter, the ordinate of the same curve is in excess on both 
sides. 

Hence, when the contact is of an even order the curves cut, and 
when of an odd order, touch without cutting. 

194; If we suppose the same quantity to be independent vari- 
able in two curves which have a contact of the w!^ order, the con- 
ditions are that at the point of contact the values of 
Xj y, dx^ dy d^x, d^y 

shall be the same in the two curves. 

H. D. c. 10 



146 CONTACT OF CURVES. 

For if any function of x is independent variable, the values 
of dx, d^x . . . d"*x must be the same in both curves when the same 
value is given to x, and d^j d^y . . . d^y can be expressed in terms 
of these quantities and of the m first differential coefficients, which 
are also the same in the two curves by the conditions of contact of 
the w^^ order. 

When any of the differential coefficients have infinite values, 
the proof in Art. I9I, which depends on Taylor's theorem, fails. 
If the infinite differential coefficients are of the second or higher 
orders, the curves are of peculiar forms (as will be subsequently 
shewn) at the point in question, such that the notion of orders of 
contact is inapplicable. If however the first differential coefficients 
are infinite, there may be no peculiarity except that of the tangent 
being parallel to the direction which has been chosen for the axis 
oi y. The difficulty is at once obviated by equating the differential 
coefficients of x with respect to y in the two curves, instead of those 
of y with respect to x, since these conditions might have been ob- 
tained exactly as the others. 

When the differentials are equated, no difficulty can arise, since 
the conditions of contact in this form might have been obtained from 
either of the other systems. 

For this reason, and also for the sake of symmetry, it is often 
better to employ the differentials than the differential coefficients. 

195. To determine the particular curve of a given group which 
has the closest possible contact with a given curve. 

Let y =f{oc) be the equation to any curve 
y' = F{x, a„ a^ . . . a„), 
that to a group of curves having n parameters kx^ a^.. .a„, by assign- 
ing particular values to which, any curve of the group may be 
determined. 

Let now such values be given to a^ «2 . . . a„, as to satisfy the 
w equations, 

y y' dx~ dx'"" dx^' ~ dor-' ^^^' 

when a particular value, a, is given to x. This can evidently be done, 
since a, Og,. . .«„ enter into the values of y' and its differential co- 
efficients ; the n parameters will then be absolutely determined by 



CONTACT OF CURVES. 147 

the assumed equations and cannot be made to satisfy any additional 
conditions. When these values are given to the parameters, the par- 
ticular curve thus selected from the group will have a contact of the 
(n - .1)*^ order with the curve j/ =/{x) at the point whose abscissa is a. 

Hence the order of the closest contact which a curve of a given 
species can be made to have with a given curve at any point of it, is 
equal to the number of parameters in the general equation to curves 
of that species, diminished by unity. 

It may happen in particular cases, that the same values of the 
parameters which satisfy the n equations (1) will also make some of 
the higher differential coefficients equal in the two curves, and in 
that case the contact will be of a higher order ; but in general the 
order of closest contact is less by unity than the number of parameters. 

The general equation to the circle contains three parameters ; a 
circle may therefore always be found which shall have a contact of 
the second order at any point of any given curve. 

196. The curvature of a curve at any point may be conveniently 
measured by determining the radius of the circle which has the 
closest possible contact with the curve at that point. Hence the fol- 
lowing definition. 

The circle which has a contact of the second order with a curve 
at any point, is called the circle of curvature of that point, and its 
radius the radius of curvature, 

197v To determine the radius of curvature and the co-ordinates 
of the centre of the circle of curvature at a point (x, y) of a given 
curve. 

Let jO be the radius, a, y5, the co-ordinates of the centre. Then 
the equation to the circle is 

(^-a)^+(^,-/3y = / (1), 

whence {x — d)dx-^ {y- P)dy =0 (2), 

(x-a)d'x + {t/-l3)d'y = -dx^-df = -ds' (3); 

where dx, dy, d^x, d^y, ds are the differentials of the co-ordinates of 
the circle. 

We have now to give to a, jS, and p such values, that the circle 
and curve may have a contact of the second order at the point {x, y), 
that is, such values that x, y, dx, dy, d^x, d^y may have the same 
values in the circle as in the curve. If therefore we give to the 

10—2 



148 Contact of curves; 

difFerentials in equations (l), (2), (3) their values derived from the 
equation to the curve, those three equations will give the required 
values of a, /3, and p. To find these explicitly, we have from (2) 

^ x — a y — (i (^~ ^) ^^^ + {y — /^) ^y 



dy 




•dx 




dyd'x- 


- dx d^y 










~ds' 




by(3) 




-dyd 


'x-dxd'y' 


x~a 


^y 


-/3 


_s/{(^-«7 + 


{y- 


-m 



dy - dx J{dx^ + dy^) 

Equating these values, we obtain 

d^^ 



X + 



dy^x — dxd?y' 
dyds^ 



l^-y- 



dyd^x — dxd^y^ 

dxds^ 
dyd^x — dxd^y' 



The magnitude of p without regard to sign being the object of 
investigation, we must employ the upper or lower sign in any par- 
ticular curve according as the former or the latter renders the above 
expression positive. The direction in which p is to be drawn will 
be determined by the values and signs of the expressions for a 
and/3. 

* The artifice of elimination here used, depends on the following algebraical 

proposition. If 

a _ c 

b~d' 

, a c ma + nc ^ ,, , . , 

t hen — = - = — J , for all values of m and », 

b d mb + nd^ 

For ma-\-nc=:-^\mb ■>f-7i- c\-'T{m,b + nd), 

and V(a2 + c2)=| ^ (6^ + |Jc2)=f- V (6' + cf^), 

■whence the above equations follow. 

By giving convenient values to m and n, as in the instance in the text, this often 
affords a ready method of elimination. 



CONTACT OF CURVES. 149 

198. If we denote by t the angle which the tangent to the 
curve makes with the axis of a:, we have 

T = tan-' -^- ; 

(fydx — d^xdy 
•*• '^''" ' dy' + dx' ' 
d-T _ dy d^x — dx d^y 
•*• d^" d? ' 

or using the lower sign in the value of p, 

ds 
P = d^' 

199. In the expressions above found for p, a, /3, the independent 
variable is general ; they may be simplified in particular cases by 
assuming different independent variables. The following are the 
most important forms and should be remembered. In the first three 
eases the upper sign of the expression for jo in Art. 197 is used, in 
the fourth the lower. These are the signs commonly used in the 
respective cases, but for the reason above given, it is perfectly im-i 
material which sign we adopt. 

(1) X independent variable. 

Putting d^x = 0, the expressions become 



{■'(l)T 



. ds' 

^ dxd^y d^y 

d? 



dy ds^ dy 



-© 



dxd^y dx d^y 

dx' 



^ ^ Vdx/ 



ds^ '^ 



dx' 

Exactly analogous equations may be obtained when y is inde- 
pendent variable. 

(2) s independent variable. 

Since the expressions do not involve d's they will still be correct 
in their general form, but it is often convenient to modify the ex- 
pression for p as follows. 



150 CONTACT OP CURVES. 



We have {dyd^x^dxd^'yY = % (1); 

also dx^ + dy^ = ds^ ; 

/. dxd^x + dyd^y — 0, since ds is constant. 
Adding the square of this equation to (1), we have 
i^xy {d^- + df) + {d'yY (d^ + dy') = ^° ; 
d^ 1 



The expression for p may also be put in the following forms, 
when s is independent variable. 

As before, dxd^x + dy d^y = ; 

ds^ 



' ' -^-dxd^y 

dx 
ds dx _ ds 

ds^ 
In like manner, by eliminating d!'y we obtain 

dy 
ds dy ds 

d^ 

(3) Q independent variable, where 

x = r cos ^, ^ = '' sin d. 
Then dx= dr cos 6 — dOr sin 0, 

d^x = d^r cos a - 2 dr dd sin d - dd^r cos 0, 
dy = dr sin + d0r cos 0, 
d^y = d'r sin + 2 Jr J^ cos - jaV sin ; 
.-. dy d'x - dx ^y = rd'r dd - 2dr^ d0 - r^d0^, 
and £/5'={rfr»+r^(;a^}t; 



CONTACT OF CURVES. 151 

{ dr'+r'dd'}^ 

P~r(Prdd-2dr'dd-r'dd'' 

drV]^ 



\ddj 



^ dd' 

(4) It is often convenient to express f) in terms of p, the per- 
pendicular upon the tangent. From Art. 188, we have, 

_a:dJ^ — ^ dx 
^~~^~~ds ' 

•'• ^p~'T~3 {{^ ^y ^ y ^^^) ^^^ "" ^* ^^ ^^^y—y ^^ \ 

.*. d'p ds^ = {(x d^y —y d^x) (dx^+ dy^) + (ydx-~x dy) {dx d^x + dy d^y)] 

= X {dx^ d^y — dy dx d^x] + y {dx dy d^y - d^x dy^} 

= (x dx-k-y dy) [dx d^y — dy d^x] ; 

— ds^ xdx -^ y dy 

dy d^x — dx d^y dp ' 

or P^~d^' '^ x'+y'=^r\ 

the negative sigi\ of the expression for p in Art. 197, being used in 
this instance in order to make the expression in terms of p and 
r positive. 

Cor. a chord of the circle of curvature drawn through the 
point of contact is called a chord of curvature. Hence the chord of 
curvature through the pole of a curve referred to polar co-ordinates, 
equals the diameter of curvature multiplied by the cosine of the 
angle between the radius of curvature and the radius vector ; 

dv t) df 

.*. Chord of curvature through pole = 2r — - = 2p -y- . 

200. Since the circle of curvature has a contact of an even order 
with the curve, it will generally cut the curve. If however the 
values of a, /^, and p^ which satisfy the conditions of contact of the 
second order, happen to be such as to make the third differential co- 
efficients identical in the curve and circle, the contact will be of the 
third order, and the circle will touch without cutting the curve. 

This will be the case whenever the radius of curvature attains a 



152 CONTACT OF CURVES. 

maximum or minimum value. For at such a point, making x inde- 
pendent variable, y =/W being the equation to the curve. 






•••o = 3/'W{/"WP-/"'(4{i+/W?} (!)• 

Now if we differentiate three times the equation to the circle, 
with X as independent variable, we have 

••• (^-«)+|(^-^)=o, 
^" i + M^V' 



or 



(S 



- . 3/(^){/"W l' 

by the conditions of contact of the second order ; 

.-. g=/"'W, by (I); 

or when the radius of curvature is a maximum or minimum, the con- 
tact is of the third order, and the circle does not cut the curve. 
This will always be the case at the vertex of a curve. 

201. To find the equation to the e volute of a given curve. 

The co-ordinates (a, /3) of the centre of the circle of curvature at 
any point {x, y) of a given curve have been found, in Art. 197^ as 
functions of x and y. As the point of contact varies, the position o^ 
the centre of curvature will also vary, and if x, y vary continuously, 
the centre of curvature will trace out a curve. This curve, for 
reasons which will be explained below, is called the evolute, the 
given curve receiving the correlative title of involute. 



CONTACT OF CURVES. 153 

The equation to the evolute will be found by eliminating x and 
^ between the equation to the involute and those which give the 
values of a and /S. 

The elimination is generally difficult, and, with a few exceptions, 
impracticable. 

202. To prove the properties of the evolute from which it 
derives its name. 

In determining the values of a, ft, p, in Art. 197? we differen- 
tiated the equation to the circle of curvature twice, and substituted 
in the result, for x, y and their differentials of the first and second 
orders, their values derived from the equation to the given curve, 
the equality of these differentials being the condition of contact of 
the second order between the curve and the circle. 

In what follows, x and y will be considered as co-ordinates of any 
point of the involute, and a, /5 those of the corresponding point of 
the evolute ; so that /o, a, /? are functions of x and y^ and no longer 
constants, as when x and y were considered as current co-ordinates 
of the circle and not of the curve. With this understanding we may 
proceed as follows to determine the properties of the evolute. 

The values of /o, a, /3 are given by the equations (Art. 197.) 

{x-ay^.{y-^y^p^ (1), 

{x-a)dx + {y-ft)dy = 0..... (2), 

{x-a)d'x + {y-^(3)d'y = -ds\.... (3). 

IDifferentiating (1) and (2), considering a, /?, p as variables, we 
obtain 

{x — a)dx + (y — (3) dy—{x — a) da — {x — p) d(3 = p dp, 

{x - a) d^x + (t/- fi) d-y ~dxda-dy d(3= - ds^, 
from which equations, together with (2) and (3), we have 

{x - a)da + {y - l3) d(3 = - p dp (4), " 

dxda+dyd/3 = (5). 

Equation (2) is the equation to the normal to the involute, a, /3 
being the current co-ordinates. Hence any point of the evolute is 
in the normal to the corresponding point of the involute. 

Again, from equations (2) and (5), we obtain : 

a;- a y-f3 

^-d^'^'W ■ ^^^' 



154 



CONTACT OF CURVES. 



'i^ by (4). 



do 



by (1). 



This is the equation to the tangent to the evolute at a point (a, jS), 
X, y being the current co-ordinates. 

Hence any point of the involute is in the tangent to the cor- 
responding point of the evolute. The radius of curvature, therefore, 
which is the line joining corresponding points of the involute and 
evolute, is at the same time a normal to the former and a tangent to 
the latter. 

If we denote by <r the arc of the evolute, we have, from equa- 
tion (6), 

X — a _y — 13 

_ (x-a)da + {y-(3)d3 
da'+d/S' 

J(da' + dl3') 

Equating these values, we have 

dp =fc dor S3 0, 

If <r is so measured that it increases as p decreases, we must take 
the upper sign, whence 

p + (r = C; 

that is, the arc of the evolute added to the length of the radius of 
curvature is invariable. 

From these properties it is easily seen that the involute must be 
a curve traced by the end of a string unwound from the evolute, 
and it is from this property that the name is derived. 

Thus, let a point P of the string pP in unwinding from the 
curve pqA trace out the curve PQ, 
and let pP, qQ he any two posi- 
tions of the string; pP and qQ 
must evidently be tangents to Aqp 
at p and q» 

Again, the portion of PQ in 
the immediate neighbourhood of 
P, may ultimately be considered 
as traced by the revolution of pP 
about p, and therefore coincides 
with a circle whose centre is p 
and radius pP. 




CONTACT OP CURVES. 155 

Pp is therefore a normal at P and p is the centre of curvature 
of PQ at P. So Qq is a tangent to Aq at g and a normal to PQ at 
Q, and gr is the centre of curvature of PQ, at Q. The curve pq is 
therefore the locus of the centres of curvature of PQ. Again, if J. 
be a fixed point in Aqp, Qq evidently exceeds Pp by the length of 
string unwound from qp ; 

.-. Aq+qQ = Ap+pP, 
which is the property expressed by the equation 

p + a- = C, 

This latter property is a necessary consequence of the former one, 
viz. that the tangent to the one curve is a normal to the other, and 
was in fact deduced from it in the last article. 

If P'Q' be a curve traced out by any other point, P', of the string, 
the same properties will belong to P'Q' as to PQ, and both of them 
will be involutes of Apq. 

Hence it follows, that while for a given involute, there can be 
but one evolute, an indefinite number of involutes can be obtained 
from a given evolute, by varying the length of the string unwound. 

203. To find the radius of curvature at a point of an evolute 
corresponding to a given point of any curve or involute. 

Let /) be the radius of curvature at any point of the given curve; 
/)' that at the corresponding point of the evolute. Then if t, t', be 
the angles made with the axis of x by tangents at the corresponding 
points of the involute and evolute, , 

since the tangent to the evolute is a normal to the involute. 

ds 
Also (Art. 198) ^"^• 

And similarly, p' = -j—, 

ClT 

_dp 

since dnr' — dT and d<y = dp'3 

, _dp ds _ dp 
" " ds' dr " ds' 

If a second evolute be drawn to the former as involute, and p"hQ 



1156 



CONTACT OF CURVES. 



the radius of curvature at the corresponding point, we shall have 
in like manner 

"'%■ 

And if />''', />". . ./o<"^ be the radii of curvature at corresponding points 
of a succession of evolutes, we shall have in like manner 



'' =''•3 



df," 



An) ^ (n-1) ^ 

by which equations the radii of curvature may be successively deter- 
mined. 

204. To find the equation to the evolute of a polar curve. 

Let PO be the radius of curvature of the curve AP ; SP = r : 




^F = p, <SF' = p', pei^endiculars to the tangent and normal at P, 
SQ = r'. Then 

OP^r'-L^ P. = p', 

.-. p^ + p'^ = r^ (1), 



r'^ = r^ + r- 



fdr\ ^ dr 



From the equation to the curve in terms of p and r, and these 
two equations, r and p may be eliminated, leaving an equation be- 
tween p' and /, which is the equati(m to the evolute. 

For examples on the present Chapter, see Gregory's Examples, 
Chapter xii. 



CHAPTER XL 



SINGULAR POINTS. 



205. Certain points of curves present peculiarities of different 
kinds inherent in the curves themselves, and not dependent on the 
position of the co-ordinate axes. Such points are called singular 
points: the most important are multiple points, conjugate points, 
cusps, and points of inflexion. 

Mtdtiple points are those through which several branches of a 
curve pass. 

Conjugate points are isolated points through which no continuous 
branch passes. 

Cusps are points at which the curve suddenly stops and returns 
in the opposite direction. At a cusp, therefore, the curve has two 
branches with a common tangent. 

Cusps are of two kinds : 

(1) When the two branches are on opposite sides of the tangent. 

(2) When they are on the same side of it. 

The two kinds are represented in the following figures. 





Points of inflexion have already been defined in Art. 184. 

In the following investigations of the analytical properties of 
singular points, the equations to the curves are supposed to be alge- 
braical and not transcendental or circular. 

206. If a straight line is drawn intersecting any curve, it will 
meet it in a number of points which 
may equal but cannot exceed the order 
of the curve. (Hymers* Conic Sections, 
Sect. 238). 

If we suppose the line to move 
parallel to itself, the number of points 
in which it meets the curve may in the 
course of its motion undergo various 
changes. Thus in the accompanying 




158 



SINGULAR POINTS. 



figure the line MPp meets the curve in two points. When by- 
moving parallel to itself it reaches either the point A where it be- 
comes a tangent or the multiple point B, the two points of intersection 
coincide. Beyond B again, the double intersection is restored. 
So if there had been more than two branches passing through B a 
greater number of intersections would have been reduced to a single 
one. 

This property will furnish an analytical test of the existence 
of a multiple point. For convenience, suppose the intersecting 
line to be parallel to the axis of i/. The ordinates MP, Mj) of 
the points of intersection for any assigned value of a: will then be 
the roots of the equation to the curve^, when solved with respect to t/. 
Whenever two of these roots become equal the corresponding point 
must either be a multiple point as at B or must have its tangent 
parallel to the axis of ^ as at A, 

To distinguish multiple points from such points of contact, sup- 
pose another set of intersecting lines to be drawn parallel to the axis 
of X. Then the points of intersection for any assigned value of 1/ 
will be given by the roots of the equation when solved with respect 
to a;; and as before, when any two of these become equal there 
must either be a multiple point or a point whose tangent is parallel 
to the axis of x. 

If therefore for any values of .r and 1/ two or more roots of the 
equation become equal both when solved with respect to x and with 
respect to 2/, the corresponding point must be a multiple point, 
since the same tangent cannot be parallel to both axes at once. 
Hence the following proposition. 

207. To determine the analytical properties of a multiple point, 
and to find the number and direction of the branches which pass 
through it. 

Let /(^, j/)-0 (1), 

be the equation to a curve cleared of radicals; x,, ^1 co-ordinates 
of a multiple point. 

If in (1) we put x = Xi, the equation must have two or more 
roots each equal to 1/1; and if we put 3/=^i, it must have two 
or more roots each equal to x^. Since the equation is in a rational 



SINGULAK POINTS. 



159 



form, the condition of its having equal roots is that the derived 
equation shall have a root of the same value*. The equations, 

/W = o, /(y) = o, 

will therefore be satisfied as well as equation (1) by those values 
of X and 1/ which correspond to a multiple point. 

To determine the values of -^ at the point in question, we have, 
by differentiating (1), 

/'(■-)+/' (3') |-0 (2), 

which gives only an indeterminate value o£~ when the conditions 

of multiplicity are satisfied. 
Differentiating (2), we have 

/"(^)+2/"(x,3,)|+/'(5r)(|J+/'(y)g = (3), 

and putting x = a;^, 1/ = t/i, 

/"(^.) + 2/"(x.,5,,)|^+/"(y,)(|;y=o (4). 

If an^ of the coefficients in this equation are finite, it will give 

two values of — -; if these are real, two branches of the curve pass 

through the point (:r, , 1/1) at the inclinations determined by the two 

values of — ; if they are impossible, no branch passes through the 

point, which is therefore a conjugate point, since its co-ordinates 
satisfy equation (1). 

If all the coefficients in (4) vanish, the values of the differential 
coefficient remain indeterminate. 

To determine them we must differentiate (3), and in the result 
make x and 1/ equal to Xi and 1/1 : to simplify the process we may 
observe that (4) might have been obtained from (2) by differentiating 

it as if -^ were constant, since the term involving -7-^ must be mul- 
tiplied by a coefficient which vanishes when x, y are put equal to 

* See Hymers* Theory of Equations, Sect. IV. It will be observed that the 
proof of the property employed in the text depends upon the equation being cleared 
of radicals. 



160 SINGULAR points; 

^i> ^I'i the same will be the case with the equation derived from 
(3), which will be a cubic equation giving three values of -^ . 

If these are all real, three branches pass through the point at 
inclinations determined by these values; if only one root of the 
equation is real, there is a single branch of the curve through the 
point (a?!, ?/i). 

If this equation becomes indeterminate by its coefficients vanish- 
ing when x, y equal j;, , ^, , we must proceed as before, differen- 
tiating as if —^ were constant, until we arrive at an equation which 

iias finite coefficients. The number of real roots will indicate the 
degree of multiplicity of the point. 

If the equation for determining -^- has equal roots, two or more 

branches will touch. Points where this occurs are sometimes called 
points of osculation. 

The equation obtained after n differentiations is generally of the 

n^ degree in -j^ , but it may happen that the coefficients of some 

number (r) of the highest powers of ->— vanish while other coeffi- 
cients remain finite, thus reducing the degree of the equation to 

dx 
n — r. If, however, we had sought the values of -7—* instead of those 

of ^, these would have been the coefficients of the r lowest instead 

of the r highest powers, and there would have been r roots of the 
equation each equal to zero. In the former case, therefore, there 
must be r roots each equal to infinity, and r branches of the curve 
parallel to the axis of y, besides those determined by the other real 
roots of the equation. 

208. In this method of determining the values of -~ it has 

72 

been assumed that -7-^ vanishes when multiplied by a coefficient 

equal to zero; this may not be true, however, when the second 
differential coefficient becomes infinite at the point in question, a 
condition which, as will be seen, generally indicates some other 



SINGULAR POINTS. 161 

peculiarity besides multiplicity, such as the existence of a point of 
inflexion or a cusp. Any error arising from this source may be 
obviated by transferring the origin to the point which we are ex- 
amining, and finding the limiting value of " when a; and 1/ approach 

zero, which will then be identical with the differential coefficient 
at that point. 

This is more laborious than the preceding method, except when 
the multiple point is originally at the origin, and as the error 
alluded to is of rare occurrence the former method is generally 
preferable. 

209. To determine the analytical properties of cusps. 

Since a cusp is in fact a species of multiple point, it will 
appear, by the same reasoning as before, that the co-ordinates of 
such a point in a curve whose equation is /(jr, y) = must in general 
satisfy the equations 

That which distinguishes a cusp from a multiple point, is that the 
two branches of the curve lie only on one side of the point instead of 
passing through it. 

If, therefore, a» h are the co-ordinates of a cusp, and we give to 
X in the equation to the curve the values a + A, a - /z, one of these 
must give two real values of y while the other renders y impossible, 
h being taken sufficiently small. 

This can only be the case, in algebraical curves, if the equation, 
when solved with respect to y, is of the form 

y = F{x) + <p{x)(x~af, 

where m is odd and n even, since the radical has two real values 
when {x ~ a) is positive and is impossible when this quantity is 
negative. 

By repeatedly differentiating the above equation, we shall at 
length arrive at one of the differential coefficients of y which con- 
tains a term involving a negative power of (a?- a) and which there- 
fore becomes infinite when x = a. Hence the distinguishing pro- 
perty of cusps is that their co-ordinates render some of the differ- 
ential coefficients of ^ infinite. 

H. D. c. 11 



162 SINGULAR POINTS. 

There is one case to which the above reasoning does not apply, 
viz. that where the cusp is of the first kind and the tangent paral- 
lel to the axis of y^ since in this case the values a — h, a + h of a? 
will each give a single real value of y. 

Here, however, the first differential coefficient is infinite, and our 
general criterion that some of the differential coefficients must become 
infinite still holds, the other necessary conditions being those of y 
having a maximum or minimum value at the point in question, 
which have been already investigated. 

If the tangent is parallel to the axis of y and the cusp of the 
second kind, the substitution o€ a + h and a — k for x in the equa- 
tion to the curve will give respectively two and no real values of ^. 

To distinguish between the two kinds of cusps when the tan- 
gent is not parallel to the axis of y, we must observe whether the 
difference of the ordinates of the curve and the tangent at the cusp, 
corresponding to an abscissa a±A, has the same sign for both the 
values of i/, or whether it is affected by a double sign. This differ- 
ence = /(a ± A) -{/(«) =t/(a) ^}. 

If we give to k that sign which corresponds to the real values 
of ^, and expand the above expression in ascending powers of h, 
the development must contain an even root of k, since a change in 
the sign of k renders y impossible. 

If this root appears in the first term of the series the^cusp is 
of the first kind, otherwise it is of the second kind ; for in the 
former case the two values of the series have opposite signs, and 
in the latter the same sign, when k is taken so small that the sign 
of the first term is that of the whole series. 

If /"(a) is finite, the test assumes a simpler form; for then the 

difference becomes = -^f'iP' =*= ^^), which ultimately has the same 
At 

sign or signs as f"{a). If, therefore, f"{a) has a double sign the 

cusp is of the first kind, otherwise of the second. 

210. To determine the analytical properties of conjugate points. 

"Whenever the co-ordinates of a point of a curve render the first 
differential coefficient impossible, no branch of the curve can pass 
through the point, since at every point of a continuous branch the 



SINGULAR POINTS. 163 

tangent of the angle which its tangent makes with the axis of a: must 
have a real value. * 

If the equation to the curve, /(a:, y) = 0, is in a rational form, 
the value of -~ obtained from the equation 

cannot be impossible for any values of x and y, since both f'{x) 

and f'{y) must be free from radicals. If, therefore, ~ have an 

impossible value, this equation must become indeterminate, or the 
co-ordinates of the conjugate point satisfy the equations 

A^>y) = o, /W = o, f{y) = o. 

There may, however, be conjugate points in a curve without 
■J- becoming impossible, in which case the above equations are not 
satisfied. 

For the point {a^ b) will be a conjugate point if y is real when 
x-a but impossible when x-a + h or a~h^ h being taken suffi- 
ciently small. This can only occur, in algebraical curves, when the 
equation solved with respect to ^ is of the form 

y = F{x) + <p {x) {x - ay (x - cf, 

where m is odd, n even, and a less than c ; for when x = a, y = F(a) 

and when x is put equal to a^k, 

y = F{a^h) + (p{a^h) {^hy^a-c ± hf, 

which is impossible when h is sufficiently small, whether we take 

the upper or lower sign, since a — c^h may in either case be made 

negative by giving a sufficiently small value to h. 

dif , , - 

If r is equal to unity, ^ will contain a term involving {x — c)", 

whose coefficient remains finite at the point in question, and will 
therefore have an impossible value. In this case, therefore, the 
above conditions are satisfied by the equation in the rational form. 

But if r is greater than unity, the terms in -^ , involving the radi* 
cals, will all disappear when x = a^ and -^ , as well as y, will have 

a real value at the conjugate point. 

11—2 



J 61 SINGULAR POINTS. 

211. There are some other peculiarities which can occur only in 
curves whose equations involve circular or transcendental functions. 
Thus y may become impossible in such curves, on one side of a cer- 
tain point, without having double values on the other, since the 
impossibility may arise from other causes than the existence of an 
even radical in the value o'i y. These are called points d^ arret. 

The curve y = x\ogx will be found to have a point d'arret at the 
origin. 

Again, the differential coefficient may change its value abruptly, 
so that two branches of the curve meet in a point at a finite angle. 

Such points are called points saillanls. The curve y = x tan~^- has 
a point saillant at the origm. 

212. The principal results of this chapter may be summed up 
as follows. 

If /(.r, y) = is an algebraical equation in a rational form, those 
values of x and y which satisfy the equations 

correspond to points which are either multiple points, cusps, or 
conjugate points. They may in general be distinguished thus. 

At a multiple point -p has two or more real values. 

At a cusp some of the differential coefficients of y become in- 
finite, and the ordinate is generally impossible on one side of the 
point. 

At a conjugate point -p is generally impossible, and y is impos- 
sible on both sides of the point. 

Ex. To find the nature of the curve 

y"^ - axy^ + a;* = 
at the origin. Differentiating we have 

{^x^ - ay"^) + (4^/^ - 9.axy) ;p = 0. 



dx 



There is therefore a singular point at the origin. Differentiating 
nstant, we obtain 



as ir -J- were constant, we obtam 
dx 



SINGULAR POINTS. 165 

all the coefficients of which vanish when x and y are put equal to 
zero. Differentiating again as if -^ were constant, we have 

which gives when x = 0, y = 0, 

' (IT- 

There are therefore at the origin two branches parallel to the axis 
of X and one parallel to the axis of j/, indicated by the disappearance 

of the coefficient of (-^ j . 

If we solve the above equation with respect to y^ we obtain 

'which gives four real values of y when x is positive and sufficiently 
small, and no real value when x is negative. 

The branches parallel to the axis of x must therefore form a 
cusp. 

Since the tangent at the cusp is the axis of x and its apex at the 
origin, the difference between the ordinates of the curve and tan- 
gent is simply the value of y, and since the corresponding values of 
y are of opposite signs, the cusp is of the first kind. 

Additional examples will be found in Gregory's Examples, 
Chap. X. 



CHAPTER XII. 

TRACING OF CURVES. 

213. When the equation to any curve is given, we can, by 
giving a series of different values to one of the co-ordinates, deter- 
mine as many points of the curve as we please. 

By determining the values of ^ or r -y- at these points, we can 

find the direction in which the branch of the curve passes through 
them; and by the methods already investigated we can find the 
position of their asymptotes, and the nature and situation of their 
singular points. 

Having done this, we can trace the form of the curve. 

214. When the equation to the curve is of the form y -f{x\ it 
is generally sufficient to determine the position of the asymptotes, 

if any, and the values of 3/ and —- for a few points where they can 

most easily be found, when the positions of points of inflexion, maxi- 
ma or minima values of the co-ordinates, &c., readily suggest them- 
selves. When it is required to determine the position of such points 
more exactly, our previous methods must be resorted to. 

The following example will shew the method to be pursued : 

a being greater than b, 

.•.^ = *(x-«)y(iZ^). 

Since the curve is symmetrical with respect to the axis of a?, we 
may trace it with the upper sign only, and add similar branches on 
the opposite side of the axis of ^, 

.•.3, = (^-a)y(^), 

^ _ , _ . /(^-^\ f_i_ 1 _J 1\ 

''' dx~^'' ''■^ \/\ a: )\x-a^2x-h 2xj ' 
(1) To find the asymptotes not parallel to the axes. 



^=(^-«)y(^). 



TRACING OF CURVES. 



167 



r ^/, 1* 1*' \ 

=^-(''4)4G-i)i+ 

Therefore the equation to the asymptote is 

and the curve lies above or below the asymptote, according as x 
is positive or negative. 

(2) Positive values of x. 

Let a? = Oj .*. ^ = CO , or the axis of ^ is an asymptote, 
J? > < S, ^ is impossible. 




x> a, 



(3) Negative values of a;; 



CO 
CO 



r ^ = co. 



Hence the form of the curve must be that of the figure. 

The curve may be verified and more exactly determined by 
seeking the positions of the maxima and minima ordinates, and the 
points of intersection of the curve and the asymptotes. 

215. When the equation to the curve gives y only in an im- 
plicit form, we may often trace the curve by obtaining simple ap- 
proximate forms of the equation which hold when the co-ordinates 
approach zero and infinity respectively. 

Let, for example, the equation to the curve be 



168 



TRACING OF CURVES. 



(1) To find the asymptote, assume 

i/=Ax + B + — + &C., 

.-. J/' =^ V + 5A^Bx\+ 5^3 (^c + 2B') x^ + &c. 
- 2a' a^y = - 2a' Aa^ + &c., 

.-. = (A'+l)x'+ 5A*Bx* + A {5A'C+ lOA'B'- 2a')x'+8cc., 
.-. A'+1=0, .-. ^=-1, 

5A*B = 0, .-. B= 0, 

5A^C+10A'B'-2a' = 0, .'. C = -fa'. 

Therefore the expanded form of the equation to the curve is 

2 a' 



2/ 



5 X 



+ &C. 



The equation to the asymptote is therefore 

and the curve lies below or above it according as x is positive or 
negative. 

(2) Omit the term x^ in the equation, and consider whether the 
equation thus obtained is for any values of the co-ordinates an ap- 
proximate representation of the given curve. It is 

/ = 2aVy, 
or y* = 2a^x^. 
When this equation holds, the dimensions in x of each of the 
terms retained are -§-, while those of the neglected term are 5 ; if, 
therefore, we suppose x to be small, the term omitted is small com- 
pared with those retained; and the equation so obtained is there- 
fore, when X is small, an approximate form of the original equation. 
Hence the given curve has a branch which approximates, when 
near the origin, to the form of the curve 

f = ^j2ax, 
which is represented in the accompany- 
ing figure. 

(3) Omit the term f. 
Then the equation becomes 

2a Vj^ = x\ 



i/ = 



2a-" 




TRACING OF CURVES. 



169 



Here the dimensions in x of the terms 
retained are 5, and those of the term neg- 
lected 15. The above equation is therefore 
approximately true when x is small. Its 
form is that of the figure. 

By omitting Sla^x^y, we should only obtain the equation to the 
asymptote, which we have already found. This would give the 
asymptote accurately in the present example ; but in general, when 
the asymptote does not pass through the origin, it would give only 
an approximate form of it, namely, a line through the origin parallel 
to it ; the former method of determining the asymptote is therefore 
in general necessary. 

From the above data we can construct the curve, which must be 
of the form represented. 




This may be verified by seeking the position of the points of in- 
flexion and the maxima and minima values of the ordinates. 

It may be further verified by examining the nature of the mul- 
tiple point at the origin, which we will now do. 

y^ - 2a Vj/ + a;' = (1 ), 

.-. {^x'-^.a'xy) + (5/- 2aV) ^ = (2). 

There will be a multiple point when, together with equation (1), 

the equations 

5x* — ^a^xy = 0, 

5/-2aV = 0, 

are satisfied. The only values of x and y which satisfy these three, 
equations are j; = 0, ^ = 0. 

Differentiating (2) as if -J- were constant. 



170 TRACING OF CURVES. 

which becomes indeterminate when x and y are put equal to zero. 
Differentiating again, after dividing by 4, we obtain 

which becomes^ when x-0, y-O, 

ax 
There is therefore one branch parallel to the axis of x, and two 
parallel to that of y, corresponding to the two infinite roots due to 
the disappearance of the two highest powers of the cubic. This 
result accords with the figure. 

216. When the equation to the curve is expressed in polar 
co-ordinates, the direction of the curve at any point is determined 
by the inclination (^) of the tangent to the radius vector where 

tan = r -7-. It is also necessary to find the polar subtangent in 

order to determine the position of the asymptotes. 

Let, for example, the equation be 

r = -y e being greater than unity; 

differentiating the logarithm of r, we have 

dr _ e sin ^ 1 + e cos 6 

7dd ~ 1 + e cos ' * * ^^^~ esin d ' 

ST=rtan<p= "" 



e sin V 

i a 

Let a = oJ ^ 1 + e' 

[ tan = 00 , 
6 > < a, r is +, 

a being the smallest value of cos~^ f — j , and therefore between J tt 

and TT ; 




e sin a + J(e^- 1) ' 



TRACING OF CURVES. 



171 



> a < TT, r IS 




e sin a —Ji^-'^)' 
^>27r-a<27r, r is +, 

[ tan ^ = CO . 

When = 0' ± 2w TT, the value of r is the same as when 6 = 0'. 
Hence all portions of the curve corresponding to values of 0, either 
negative or greater than Stt, only repeat those already found between 
the limits and Stt. 

By observing the rule with respect to the sign of ST^ the asymp- 
totes corresponding to = a, and = 2'7r— a respectively, will be 
Pp, Qq, and the curve will be represented by the figure. 




A being the point where = and a that where = Stt. The 
asymptotes evidently intersect at C the bisection of Aa, since 



ST=^ST = 



and 






ST 
sin a 



Additional examples will be found in Gregory's Examples, 
Chapter xi. 



CHAPTER XIII. 

ENVELOPES, DIFFERENTIALS OF AREAS, ETC. 

217. To find the equation to the envelope of a group of curves, 

Let A^^.a) = 0.. (1), 

be the equation to the curve whose parameter is (a), 

/(x,y, a + Afl) = (2), 

that to another curve of the same group, whose parameter is 

(a + Afir). 

These curves will intersect in the points whose co-ordinates are 
given in terms of a and Aa, by equations (1) and (2). As Aa varies, 
this point of intersection changes its position, and the limiting posi- 
tion to which it approaches when Aa approaches zero is called the 
point of ultimate intersection of the curve (l), and the contiguous 
curve of the group. At the point of intersection of (1) and (2) we 
shall have 

/{x, y, a + Ag) -f{xy y, «) _ ^ . 
Aa ""' 






and, therefore, at the point of ultimate intersection, 

Aa 
... ^/^(wO^o (3). 

Equations (l) and (3) therefore determine the point. 

By eliminating a between (1) and (3), we obtain an equation 
between x and y^ which holds at the point of ultimate intersection, 
and which is independent of a. This is therefore the equation to 
the locus of the points of ultimate intersection of curves of the group 
represented by equation (1). 

Let the equation to this locus be 

0(^,5/) = O (4). 

Then it may be shewn that the curve (4) touches each of the 
curves of the group (1), and that every point of (4) is touched by 
some curve of the group. For this purpose it is only necessary to 



ENVELOPES DIFFERENTIALS OF AREAS, ETC. 173 

bear in mind that any one of the equations (I) (3) and (4) is deriv- 
able from the other two. 

Thus, let a have a particular value in (1) so that that equation 
may represent some one curve of the group. Then at the point of 
intersection of (1) and (4) equations (1) and (4) hold together, and 
therefore at that point equation (3) also holds. 

Now the value of -j- at the point of intersection, in the curve (l) 
is given by the equation 

/'W+/'w|=o (5). 

and that in the curve (4) by the pair of equations 

/'W+/w|+/'«|=ol (6). 



which are identical with (5), and therefore the curve (4) touches the 
particular curve (1). 

So also, if we take any point of (4) whose co-ordinates are 
(j:,, ^i), it will meet that curve of the group whose parameter is 
given by equation (1 ) when x, , y^ are put for x, y. 

The parameter thus obtained will therefore also satisfy equation 

(3)with the same values of a?, ^ ; and the values of ~- at the point of 

contact will be determined by equations (5) and (6) respectively, 
which as before are identical. Hence, any point x^, y^ of the curve 
is touched by one of the curves of the group (1). 

For these reasons the locus of ultimate intersections of a group 
of curves is also called the envelope of the group. 

218. If the equation to the group contains two or more parame- 
ters connected by equations, so that one only is independent, it may 
be reduced to the assumed form by eliminating all the parameters 
except one, by means of the equations connecting them. 

It is however often more convenient to differentiate the equation 
before such elimination, and afterwards eliminate the differential 
coefficients of the dependent parameters by means of the given 
equations between them. The operations in this case become more 



174 ENVELOPES DIFFERENTIALS OF AREAS, ETC. 

symmetrical by the use of difterentials instead of differential coeffi- 
cients, when the elimination may generally be most readily effected 
by the method of indeterminate multipliers, explained in Art. 177' 

As an example, we will find the envelope of the system of lines 
of a constant length whose extremities lie in two lines at right 
angles to each other. 

Let the two lines be taken for the axes of x and y ; c the constant 
length of each line of the group. 

The general equation to any line of the group is 

M- (»' 

where a, b are parameters connected by the equation 

a' + b' = (f (2). 

The equation to the envelope will be given by the elimination of 
a and b, between (1), (2), and the equations 
jrda ydb _ 
a b^ 

ada + bdb = 0, 
which together are equivalent to equation (3) of Art. 217. 
Using an indeterminate multiplier \ we have 

(^ - \a\ da^H-^- \b\ db = 0, 

whence we may obtain 

and 1^ - a6 = 0, 




b 
therefore substituting in (1), the required equation is 

For examples see Gregory's Examples ^ Chapter xni. 



ENVELOPES DIFFERENTIALS OP AREAS, ETC. 175 

219. To find the point of ultimate intersection of a normal to a 
given curve with the contiguous normal, and the locus of all such 
points. 

Let the equation to the curve be 

5'=/W (!)• 

Let x\ y' be the co-ordinates of the point of ultimate intersection 
of the normal at (a?, y) with the contiguous normal, and let the dis- 
tance between (jb, y) and (a/, y') be denoted by /j. Then 

(y-x)^+(y-^y=/ (2). 

The equation to the normal is 

{x'-x)dx + {y'-y)dy = (3). 

And at the point of ultimate intersection the differential of this equa- 
tion with respect to the parameters x, y, must also vanish ; 

.-. {x'-x)d'x + {y'-y)d''y = dx^ + dy^ (4). 

Equations (2), (3) and (4) will give x', y' and p in terms of x 
and y and their differentials, y and the differentials may then be 
eliminated by means of equation (1) and its derived equation, and 
j/, y' and p obtained in terms of x alone. By eliminating x between 
the values of a?' and y' we shall obtain the locus required. 

The equations for determining x\ y' and p are the same as those 
found in Art. 197 for the determination of a, /3 and p. 

The point of ultimate intersection of contiguous normals is there- 
fore the centre of curvature, and the locus of all such points the 
evolute of the given curve, a result which might have been foreseen 
from geometrical considerations. 

220. This property furnishes a more convenient method of ex- 
pressing p in terms of ^ and r than the one given in Art. 199* 

Let P be any point of a curve, 

>S'P = r, SY='p, \p 

O the centre of curvature at P, and therefore 



FO^p, 




0/ 



Then 
SG" = SP^ + P(J-2 SF. PO cos SPO 

= r^+ p^-2pp. S 

Since is the point of ultimate intersection of contiguous nor- 
mals, a change in the position of P will in the limit produce no 



176 



ENVELOPES DIFFERENTIALS OF AREAS, ETC. 



Jt£ 



JV 



change in the position of or the length of OP. The above equa- 
tion may therefore be differentiated with respect to r and p, con- 
sidering SO and p as constant. Hence 

= 2rdr-2pdp; 
dr 
''P = 'Tp' 

221. To find the differential of the area included by a curve 
and two ordinates. 

Let PQ be two points of a curve whose co-ordinates are AM = Xj 
MP = y, and AN=x + Aar, NQ, = y+£^y; 
and let the area between AM^ MP, the 
curve CP3 and some fixed ordinate equal A . 

Then the area MPQN will with the 
same notation equal AAj AA being the 
increment of A corresponding to an in- 
crement A^ of X. 

This lies between the areas of the pa- 
rallelograms MR, MQ, i. e. between yAx, and (y + Ay) Ax ; and 
since the limiting ratio of these areas when Ax approaches zero is 
unity, that of AA and either of them must equal unity, 

or dA =ydx. 

222. To find the differential of the area included by a curve 
and two radii vectores. 

Let P, Q be points of a curve whose polar co-ordinates are r, 0, 
and r + Ar, 6 + Ad, and let A be 
the area between the curve CP, 
the radius SP, and some other 
fixed radius. Then will SPQ, equal 
AA, the increment of A corre- 
sponding to an increment A^ of 6. 
AA lies between the areas of the 
circular sectors whose radii are SP, SQ respectively, and vertical 
angle equal to PSQ, i. e. between 

'IaB and ^I^^A0: 
2 2 




ENVELOPES DIFFERENTIALS OP AREAS, ETC. 



177 



and since the limiting ratio of these sectors when A 6 approaches 
zero is unity, that of AA and either of them must equal unity ; 

AA 



ltA=. 



^^0 ^2 



= 1, 



Ad 



.'. dA = -dd. 

2 

223. To find the differential of the volume included by a sur- 
face of revolution and two planes perpendicular to its axis. 

Let V be the volume produced by the revolution round AN 
of the area bounded by MP, the curve CP and a fixed ordinate, 
(see fig. Art. 221). Then AV will be the volume produced by the 
revolution of VQ, A F being the increment of V corresponding to an 
increment A^ of .r ; AF lies between the cylinders produced by the 
revolution of PR^ QS respectively, i. e. between 

'K'lfAx and tt (?/ + Ayf Ax : 
and sincQpthe limiting ratio of these volumes when Ax approaches 
zero is unity, that of AFand either of them must also equal unity; 

AV 



It 



A^O 



mifAx 

.'. dV='iry^dx. 



224. To find the differential of the surface of a solid of revolu- 
tion bounded by two planes perpendicular to its axis. 

Let S be the surface produced by the revolution of a curve CP 
about the axis of x, AM. Draw Pn, 
Qm parallel to the axis of x and each 
equal to PQ. Then AS, the incre- 
ment of S corresponding to an incre- 
ment Ax of X, will be the surface pro- 
duced by the revolution of PQ, and J ^f j^" 
PQ = A* if arc CP = s ; AS lies betwen the surfaces produced by the 
revolution of Qm and Pn, i. e. between ^iryAs and 27r(y + Ay) As. 
Since the limiting ratio of these surfaces when A* approaches ?ero is 
unity, that of AS and either of them must equal unity ; 

AS_ _ 
•*• ^^--'IVyAs'^' 

.'. dS = 2'Kyd^, 

^^tryj^dx^' + dy'). 
H.D C. 12 



CHAPTER XIV. 

INTEGRATION BETWEEN LIMITS AND SUCCESSIVE 
INTEGRATION. 

225. To express an integral as the limit of the sum of a series. 
Let /'(a:) be any function of x, a any constant, and let the interval 
^ - « be divided into n parts each equal to Ao;, so that 

Aa? = . and .*. x = a + n Aor. 

n 

Let (p {x) =f{a) Ao? +/(« + Ajc) A^ +/(« + 2 Aa?) Aa? 

+/{a + («-l)Aa?}Aar. 

The value of <p (x) depends on the number of terms (w) of this 
series, and the consequent magnitude of Ar. Now when w ap- 
proaches infinity, A.r and therefore each term of the above series 
approaches zero, and (p {pc), whose actual value then becomes indeter- 
minate, will approach a certain limiting value ; let this be \/^ (x). 

To determine the value of \/^ (£) change x into x + A^ in the 
above series, which is equivalent to adding to it the term 

f{a + n Ax) Ax J or f(x) Ax. 

Hence, (p {x + Ax) - (p (x) ^f\x) Ax ; 

(p {x + Ax) - (p (x) ,f . 
or ^^ ^ =f{x). 

In this equation let Ax approach zero; then <p{x) becomes in 
the limit >/^(^), 

therefore -v/zCj:) must be one of the values contained in the general 
Integral, /(a;) + C, o£/'(x). To determine the value to be given to C, 
we have 

^k(ix)=f{x) + a 



INTEGRATION BETWEEN LIMITS. 179 

since cp (x) and therefore y^ {x) evidently vanishes when x is made 
equal to a ; 

/. C = -/(«), 

and >/-W=/W-/(«) (1). 

The general form of the integral of f'{x), \iz, f(x) + C, is 

"written if(x)dx, the indeterminate constant being included in the 

integral sign. When such a value is given to C that the integral 
shall vanish when x equals any particular value a, the integral 

f(x) -/(a) is written I f\x)dx; and if in this we give to x any 

J a 

particular value h, the resulting value of the integral, viz. 
/(S)-/(a) is written \f{x)dx, 

jf{x)dx is called the indefinite integral of /'(a:), 

\ f {x)dx the corrected integral of f (x), 

\f{x)dx the definite integral of /'(a;), between the 

limits a aixd b. 

With this notation equation (1) becomes, giving y\/{x) its value, 
and writing f{x) instead o£f'(x), 

rf(x) dx = Z^A=o {/(«) ^ +/(« + Ao?) Ax ... +f(x - Ax) Ax} ; 
.«. I f(x) dx = //a=o {/(«) Ax +f(a + Ax) Ax ... +f(b - Ax) Ax}. 

The indefinite integral must therefore equal the limit of the 
sum of any number of terms of a series formed like the above, 
commencing with any term whatever of the series, and termi^ 
nating with that in which the quantity under the functional sign 
is a? - Ax. 

The above equations may be written for the sake of brevity as 
follows : 



/; 



f(x)dx = It^^o 2 /-^M/W^}, 
f(x)dx = Ua^o^^-^^A^)^^}' 



12—2 



180 



INTEGRATION BETWEEN LIMITS. 



226. To determine the area included by a curve and two given 
ordinates. 

In Art. 221 it was proved that if A is the area included by a 
curve, a fixed ordinate, and an ordinate whose abscissa is x, 

dA = y dx^ 

,'. A= ji/ dx, 

= (p{x)-\-C suppose. 

The area and the integral are alike indeterminate, the former on 
account of the indeterminate position of the iixed ordinate, and the 
latter on account of the arbitrary constant which enters into it. 

If the fixed ordinate is made definite and taken to be that 
whose abscissa is a, the arbitrary constant must have such a value 
given to it that the area shall vanish when x is made equal to a ; 

'.'. (p{a)+C^O, 
and A=^<p{x)-^ (a), 

-/V'^^ • (1). 

By giving different values to x, we obtain the values of the areas 
bounded by the ordinate whose abscissa is a and any other. Thus 
the area included between ordinates whose abscissae are a and b 



i: 



.(2). 



"^jjdx 

It may be observed that as soon as a constant value is given to 
the abscissa of the second ordinate, the form of the function <p 
disappears from the expression, so that while from (1) we can 
obtain the value of any one of a system of such areas, (2) will 
determine only the particular area to which it is equal. 

For example, let the curve be a circle. 

p 




c jr A 

Taking C for origin, we have 



INTEGRATION BETWEEN LIMITS. 181 

/. ai-ea CBNP = j'j{a^-a;')dx 

= 4-^x/K- ^') + 4«' sin"' - + C". 
fit 

Since the area vanishes when or = 0, C must also equal zero, 

.-. area CBNP - i a:J(a' -x') + i a' siir' ^ . 

Let now x = aj 

.'. quadrant A CB = ^ a^ i ^ _ i ^ ^2^ 

.'. whole area of circle = ira^. 
From this expression the form of the general function from which 
it is derived has disappeared. 

227. To determine the length of the arc of a given curve. 
If s is the arc of a curve measured from any fixed point to the 
point whose abscissa is ocj 



■■■-iA'<m-- 



If the abscissa of the fixed point is a, 

•=lVI-(i)> 

And the arc between two points whose abscissae are a and b, 



=iV{-(l)>- 



As an example we will find the arc of the cycloid from the vertex 
to a point whose abscissa is x. 



In the cycloid | = y(?^). 



a being the radius of the generating circle, and the vertex the 
origin ; 






2 J(2ax) + C, 



182 INTEGRATION BETWEEN LIMITS. 

and C =s 0, since the arc vanishes when a; = 0, 

,'. arc = 2 J{2ax). 

If we put X = 2aj we obtain the arc of a semi-cycloid which there- 
fore = 4a. . 

228. To determine the area included by a curve and two given 
radii vectores. 

If A be the sectorial area included between a curve, a fixed 
radius and a radius inclined at an angle Q to the prime radius 

.-. A^ijr'dd. 
If the fixed radius is inclined at an angle, a, to the prime radius 

A=ijydd, 

and the area bounded by radii inclined at angles a, /? to the prime 
radius 



=»i' 



d9. 



229. In like manner if Vj S be the volume and surface of a 
solid of revolution bounded by planes whose abscissae measured 
along the axis of revolution are a and a?, 

V=ir\y^dx, Art. 223, 
If the two abscissae are a and h, 

230. These expressions for areas, arcs, &c., may be found at 
once from the considerations of Art. 225. 

Thus, to find the area ABPN. Let OA = a, ON=b, Ofn = Xy 
mp =y. Divide AN into a number of parts each equal to Ao?, and 



INTEGRATION BETWEEN LIMITS, 
JP 



183 




A m n jv 

on these as bases construct parallelograms as in the figure. 

Then, assuming that the area is equal to the limit of the sum of 
the parallelograms when Aa? approaches zero, we have 

area ABNF = U^ 2«^^"yAar, 

since ^ Aa? is evidently the area of any of the parallelograms, as np, 

.*. area ABNF = I ydx. Art. 225. 

The parallelograms or other portions into which the whole quantity 
is divided are called elements. 

By taking as our element the cylinder generated by the revolution 
of one of the parallelograms, and assuming that the volume of the 
solid of revolution is the limit of the sura of the cylinders when 
A.r approaches zero, we have 

r=/^A=o2/-'^"T/Ar, 



= TT I y^dx» 



By taking the chord pq as our element and assuming that the 
arc BP equals the limit of the sum of the chords, when Aj: 
approaches zero, we have 



' A=o ^a 



By taking as our element the surface generated by the revo- 



184 INTEGRATION BETWEEN LIMITS. 

lution of the chord pq about the axis of x, and assuming that the sur- 
face of revolution is the limit of the sum of these elements when Aj; 
approaches zero, we have 

=-/>y{-(S)i- 

By taking as our element the circular sector whose radius 




S A 

is aS/j and vertical angle A0 and assuming that the sectorial area 
BSP between radii, for which the values of Q are a and /3, is 
the limit of the sum of such elements when A0 approaches zero, 
we have 

area BSP = It^^, ^J'-''^ ^r' Ad, 



< 



r'dd. 



It will be observed that the assumptions made in each case are 
equivalent to the equations giving the differentials of the areas, 
arcs, &c., in Arts. 221—224. 

In fact it is the same thing to say that the whole quantity 
equals the limit of the sum of the elements, as that the increment of 
the whole and the element have ultimately a ratio of equality, which 
is what those equations amount to when expressed in words instead 
of symbols. 

231. This method of dividing a quantity into elements may 
be applied to other functions besides areas, volumes, &c. 

Thus let it be required to find the limit of the sum of every 
portion of a circular area, multiplied by the square of its distance 
from the centre, when the magnitude of each portion is indefi- 
nitely diminished. Let the radius of the circle be a. Take as an 
element the annulus whose inner radius is r and breadth Ar. 



INTEGRATION BETWEEN LIMITS. 185 

The required function for this annulus 
= (area of annulus) r^, 
neglecting the difference of the values of r for different parts of the 
annulus, since when the limits are taken any error from this source 
vanishes. 

Therefore the required quantity 



T^dr 



4 
= (area of circle) — . 

Successive Integration. 

232. Hitherto we have considered integration only as the con- 
verse of the simple differentiation of a function of one variable. It 
may however be extended in the same way as differentiation, and the 
converse of the operations of successive and partial differentiation 
will be called successive and partial integration. 

Such operations are denoted by merely prefixing the integral 

sign (I) to the differential, of whatever kind it may be, as many 

times as there are operations to be performed. 

Thus if u=f{x) 

we have, when x is independent variable, ' 
du =/' {x) dx, 
d'u=f"{x)dx\ 



the converse of which equations will be written, 
u = if (x) dx, 



So if u =f(x, y), 

d^u =f {x) dx, d^u =f" (a;) dx\ dj,u = f" {x, y) dx dy. 



186 INTEGRATION BETWEEN LIMITS. 

X and y being the independent variables, whence we have 
n = jf{x)da: = jjr(x) dx^ = j jr{x,y)dxdy. 

The last integral is more often written idx \dyf"{x^y), a form 

which will be found convenient when the integration is between 
definite limits. 

An integral involving two or more signs of integration is called 
a double or multiple integral. 

233. We have seen that in order to obtain the general value 

of I/' (-3?) dx it is necessary to add an arbitrary constant, because 

such a quantity would disappear in differentiation. The same must 
be done at each stage of the successive integration of a function of 
one variable. 

Thus (f"{x) dx =/(x) + C; 

••• f[/'{^)da:' = j(J'x+ C)dx=f(x) + Cx+C', 
where C and C^ are arbitrary constants. 

So if we integrate n times in succession with respect to x we 
must add, 

Cjj?"-^ + Cj,x"-' + + C„.iX + On, 

since such a quantity would disappear in the corresponding differ- 
entiation. 

For the same reason, when we integrate a partial differential 
djf(x, y) we must add the most general quantity which would have 
disappeared in obtaining d^f{x^ y). This may be any function o£y 
or a constant, both of which are considered as included in the general 
form (p (y) where <j) is perfectly arbitrary. 

If therefore ti =/(^, y), 

ff{^)dx=/(x,y) + cp{y). 



INTEGRATION BETWEEN LIMITS. 



187 



Similarly, /"/(y) dy =f{x, y)+ylr (x), 

when «^, -^ are perfectly arbitrary functions. 

So if we have to integrate twice partially with respect to x, we 
must add an arbitrary function of _^ at each stage ; 



# 



<py and <p2 being arbitrary functions. 

Again, if we have to integrate first partially with respect to one 
variable and then partially with respect to the other we have 

jdyjdxf" (^. y) =jdy {/{y) + <p (^)} 

where -v/^ (y) is the integral of the arbitrary function ^ (y) and there- 
fore itself a perfectly arbitrary function of y, and x {^) a perfectly 
arbitrary function of x. 

In like manner the general value of any multiple integral will 
be given by adding to the particular integral, such functions of the 
variables as wou^d have disappeared in the corresponding differen- 
tiation. 

234. The arbitrary functions introduced into the value of par- 
tial and successive integrals may be determined, and the definite 
integral obtained between given limits exactly as in the case of 
simple integration. 

The case of most frequent occurrence is that of successive inte- 
gration with respect to several different independent variables, as 
for instance where 



u=\dx \dyf"{x,y). 
Here jdyf^x, y) = ^-l^ + <p (x), 



where (p is arbitrary. 

If the integral vanishes when y equals some function, Xj, of x, 
we have 

^ df(x,X,) ^,, 



188 INTEGRATION BETWEEN LIMITS. 

which determines and then 



/ 



.,*/-('..)-%^-i^'' 



£ 



If both limits oiy are given and equal X^ and Xg, we have 

^F'{x) suppose, 

where F'{x) will be a function of ^ whose form depends on that of/ 
and on the values of Xj and Xj. 

dyf'ixj y) is thus expressed as a definite function oi x in the 

^1 



/. 



same 



way as I dxf{x) was in Art. 225 expressed as a definite 

constant. Substituting the above value in the general integral, 
we have 

^dx\^%yf"{x, y) = jdxF\x) 

= F{x) + a 

If this integral is to vanish when x = a, 
= F(a) + C; 

... jjxQdyfXx, y) = F (x) - F(a). 
And if two limits a and b are given, 

lldxJ2dy/"{^,y)^F{h)-F{a). 

235. The interpretation of multiple integrals as the limiting 
sums of series, is similar to that of simple integrals. Thus in the 

CX2 
above instance I dyf"{x, y) is the limiting value of the series 

JXi 

whose general term is f"(x, y) Ay, the first and last terms being 
those in which y = Xi and Xj — Ay respectively. The limit of the 
sum so found is what we have denoted by F'{x). This is a quantity 
varying with x, and a series may therefore be formed whose general 
term is F^(^x)Ax, and the first and last terms those in which x equals 



INTEGRATION BETWEEN LIMITS. 



189 



a and h — Aa; respectively. The limit of the sum of this series will 
then be 1 F'{x)dx, Hence 

that is, if an element be formed whose value is, f"{xj y)AxAy, the 
limit of the sumi of all such elements will be given by the mul- 
tiple integral \dx \dyf"{xjy\ taken between the appropriate limits. 



236. We will apply this method to find the area included be- 
tween the two curves APB, AQB, whose equations are respectively 

y=/W and y = (p{x). 




Let Xy y be the co-ordinates of any point p within the area ; 
X + Aj:, y + Ay those of any neighbouring point q ; x and y will there- 
fore be independent of one another. Then the area of the paral- 
lelogram pq whose sides are Ax and Ay = Ax Ay. 

Also the area of a parallelogram of which one side is PQ and 
the other Ax is the sum of all the smaller parallelograms obtained 
by giving to y all values between MP{f(x)} and MQ{^ (x)} ; 



.*. parallelogram PR = j dy Ax 



= {cp(x)-f(x)}Ax, 

and the whole area between the curves equals the limit of the sum 
of all parallelograms formed in the same way as PR, and comprised 
between the ordinates HA, KB, that is, between the limits a and b 
o£x,i£OH=a, OK = b. 



190 INTEGRATION BETWEEN LIMITS. 

.-. Area = j dx j dt/ = I dx {(p (x) -/(x)} 

Ja Jf\x) Ja 

= j (p(x)dx — j f(x) dx. 

We might have obtained the same result by the formula of Art. 
226, which would have given. 

Area HAQBK= j'(p(x)dx, 
. Area HAPBK= [/(j;)^^?; 

.'. Area between the curves =1 (l>(^x)dx- j f(x) dx. 

237. As another example, let it be required to find the centre 
of gravity of a quadrantal area. 

Let S, y be the co-ordinates of the centre of gravity G; x,y those 
of any point p within the area. Then the moment c p 
of G about OB must equal the limit of the sura of 
the moments of all the elements of which the area is 

made up. 

o JI A 
Or if we take an element Ax Ay as before, we have, 

area BOC xAG = U^ MpAx Ay^ 
also area BOC = ICL Ax Ay ; 

-_lt^y AxAy 
•' ^~ It^AxAy 

jdxjydy 

jdxjdy 

the limits being first from y = Otoy = MP = Ja'-x', and then from 
a: = to ;» = OB = a, a being the radius of the circle ; 




Similarly 



\ dx\ y dy 

yJl b—^ 

ra rVail^ 

I dx j xdy 

ra /•Va2_.r2 



INTEGRATION BETWEEN LIMITS. 191 

Now I ydy=-^+C; .: 0=C; 



Jo 



a^-x^; 



/■Va2_x2 /•Va2_a;2 

Also I dy = sja^ - x^, ^^^ Jo xdy = xj^ 

•••AH ^''^=3- 

Also j„rf^j„ % = jo<^7«''-^ = i'^N/«'-'=^' + i^'sm-^- + <^^ 









3' 



- 3 4a 
4 



and jc= =-^ 

4 



238. The method of converting the limiting sums of series into 
multiple integrals, is equally applicable whatever may be the number 
of independent variables. In every case, the increments are changed 
into the corresponding differentials and the sign of summation into 
that of integration, the limits of the integration being the same as 
those of the series. The order of integration is very material. For 
in general the limits at each integration are functions of all the 
variables with respect to which the integration remains to be per- 
formed ; their values therefore depend on the order of integration, 



392 INTEGRATION BETWEEN LIMITS. 

and the latter cannot be varied after the limits are once fixed. The 
only exception to this, is when all the limits are constant, as some- 
times happens, in which case we may differentiate in any order we 
please. A careful consideration of the examples above given, will 
make the truth of these observations apparent. In working ex- 
amples great care ^is requisite to avoid errors in fixing the limits, 
which can only be done with accuracy by keeping constantly in view 
the series of which the integrals are the limiting values. 

Examples of the determination of areas, surfaces, &c. will be 
found in Gregory's Examples, Part ii. chap. ix. Applications of in- 
tegration to summation, are met with in all physical subjects, and 
form the principal use of the Integral Calculus. 

239. To express an integral by an infinite series of finite terms. 
By integrating by parts, we obtain 

\f(jic) dx =/{x) X — \f{x) X dx, 
jf '{:>:) X dx =/' (x) I" - i jf"(x) X' dx, 

jf"{x) X- dx =/" (:.) J - 4 |/"'(x) ^ dx, 
&c. = &c. 

If in this series we make n infinite, we have 
\f{x)dx=f{x)x-f{x)~ +/-(x)-^-&c in inf. 

The term -p //"^ {pc) ^" dx is therefore the difference between 

the infinite series and its first n terms. An arbitrary constant must 
be added to give the general value of the integral. In the first form 
of the expression the constant is included in the last term. 

240. From the formula of integration by parts we may obtain 
a proof of Taylor's Theorem, which presents the remainder after n 
terms in the form of a definite integral. 



/(a + A)-/(a)=J"Vw^^; 



INTEGRATION BETWEEN LIMITS. 193 

let x = a + h~z; 

.'. dx = — dz; 
and when x = a, z = h, 

x = a + h, z = 0; 

.-. f(a + h) -f(a) = (/(« -^h-z) dz. 
Now by repeatedly integrating by parts, as in Art. 239, 
f{a + k-2)dz =f{a +h-z)z +f{a + h-z) "' 



/ 



^ 



-/-"(« + * - -) ]S + [^//'"'(" +* - ^) ^"'' 



Therefore taking the integral between the limits and h, 
f{a + h) =/(«) +/'(a) A +/'(a) | ...... +/i»->i(a) ^ 

The value of the remainder after n terms is, therefore, 

N T-^ [ /"> (a + h-z) z"-' dz. 

This is easily shewn to lie between the limits of the remainder 
before found. 

For /*/("> (a + k-z) z''-' dz = It^^ So""-^^ {/"> {a + h-z) s"-^ A^}. 

The particular value of y^"^ {a + h - z), by which any term of this 
series is multiplied, must lie between the greatest and least values 
w^hich that function can assume while z varies from to h. 

There must, therefore, be some intermediate value of z which 
gives to the function a value, which multiplied into every term, 
gives the same result as when each term is multiplied by the par- 
ticular value belonging to it. This value of ^ may be represented 
by (1 -^) hj where 6 lies between and 1. By substituting this, the 
series becomes 

H. D. C. 13 



194 INTEGRATION BETWEEN LIMITS. 

••• /V"^(« + h-z) z'^' dz =/<")(a + eh) Tz"-^ dz 

And the value of the remainder is, therefore, 

the expression before obtained. 

The expression in the form of a definite integral is of course 
preferable, when the integration can be effected, as it determines 
the exact value of the remainder and not merely the limits between 
which it lies. 



THE END. 






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